Worm domains and Fefferman space–time singularities

Journal of Geometry and Physics 120 (2017) 142–168

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Worm domains and Fefferman space–time singularities Elisabetta Barletta a , Sorin Dragomir a, *, Marco M. Peloso b a

Università degli Studi della Basilicata, Dipartimento di Matematica, Informatica ed Economia, Via dell’Ateneo Lucano 10, 85100 Potenza, Italy b Dipartimento di Matematica, Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy

article

info

Article history: Received 29 March 2017 Received in revised form 5 June 2017 Accepted 6 June 2017 Available online 19 June 2017 Keywords: Worm domain Levi form Fefferman’s metric Curvature singularity Schmidt metric Bundle boundary

a b s t r a c t Let W be a smoothly bounded worm domain in C2 and let A = Null(Lθ ) be the set of Leviflat points on the boundary ∂ W of W . We study the relationship between pseudohermitian geometry of the strictly pseudoconvex locus M = ∂ W \ A and the theory of space–time singularities associated to the Fefferman metric Fθ on the total space of the canonical π circle bundle S 1 → C (M) −→ M. Given any point (0, w0 ) ∈ A, we show that every lift Γ (ϕ ) ∈ C (M), 0 ≤ ϕ − log |w0 |2 < π /2, of the circle Γw0 : r = 2 cos[log |w0 |2 − ϕ] in M, runs into a curvature singularity of Fefferman’s space–time (C (M), Fθ ). We show that Σ = π −1 (Γw0 ) is a Lorentzian real surface in (C (M), Fθ ) such that the immersion ι : Σ ↪→ C (M) has a flat normal connection. Consequently, there is a natural isometric immersion j : O(Σ ) → O(C (M), Σ ) between the total spaces of the principal bundles of Lorentzian frames O(1, 1) → O(Σ ) → Σ and adapted Lorentzian frames O(1, 1) × O(2) → O(C (M), Σ ) → Σ , endowed with Schmidt metrics, descending to a map of bundle completions which maps the b-boundary of Σ into the adapted bundle boundary of C (M), ˙ ) ⊂ ∂adt C (M). i.e. j(Σ © 2017 Elsevier B.V. All rights reserved.

1. Worm domains and Fefferman’s metric

{

}

A worm domain is a smoothly bounded pseudoconvex domain W in C2 given by W = (z , w ) ∈ C2 : ρ (z , w ) < 0 where

⏐ ( ) 2 ⏐2 ρ (z , w) = ⏐z − ei log |w| ⏐ − 1 + η log |w|2 ,

(1)

and η ∈ C (R) satisfies the conditions: (i) η(t) ≥ 0 and η is even and convex; (ii) η (0) = Iµ ≡ [−µ, µ] ⊂ R; (iii) there is a > 0 such that |t | > a yields η(t) > 1; and (iv) η(t) = 1 implies η′ (t) ̸ = 0. Originally devised by K. Diederich and J.E. Fornaess [1] to produce examples of pseudoconvex domains without a Stein neighborhood basis, worm domains are known (cf. e.g. [2]) to exhibit an array of peculiar irregularity properties. Perhaps the most remarkable of such properties, is the failure of satisfying Condition R, that is the condition that the Bergman projection −1

∞

Pf (z , w ) =

∫

K (z , w, ζ , ω) f (ζ , ω) dV (ζ , ω),

(z , w ) ∈ W ,

W

maps C ∞ W to C ∞ W (cf. M. Christ, [3]). Here K (z , w, ζ , ω) is the Bergman kernel of W and dV denotes the Lebesgue measure in C2 . It is well known that any smoothly bounded strictly pseudoconvex domain satisfies Condition R (see [4]

( )

*

( )

Corresponding author. E-mail addresses: [email protected] (E. Barletta), [email protected] (S. Dragomir), [email protected] (M.M. Peloso).

http://dx.doi.org/10.1016/j.geomphys.2017.06.001 0393-0440/© 2017 Elsevier B.V. All rights reserved.

E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

143

and [5]) and indeed W is only weakly pseudoconvex, for the Levi form of its boundary vanishes at each point of the annulus A = {(0, w ) ∈ ∂ W : ⏐log |w|2 ⏐ ≤ µ}

⏐

⏐

(cf. Proposition 1.3 in [2], p. 480). The goal of this paper is to understand the nature of weakly pseudoconvex points on ∂ W by investigating the behavior of geometric objects as points approach the critical annulus A. We do this by combining methods in pseudohermitian geometry, as created by S.M. Webster [6], and the theory of space–time singularities, see. e.g. C.J.S. Clarke, [7], by profiting from the presence of the Fefferman metric of W , a Lorentzian metric defined solely on a saturated open subset of the total space of the canonical circle bundle S 1 → C (∂ W ) → ∂ W . If U(z , w, ζ ) = |ζ |2/(n+1) u(z , w ),

u(z , w ) = K (z , w, z , w )−1/(n+1) ,

(2)

with n = 2 then one may equip W × (C \ {0}) with the (0, 2)-tensor field G=

2 ∑

∂ 2U

∂ zA ∂ zB A,B=0

dz A ⊙ dz ,

z0 = ζ ,

B

z1 = z,

z 2 = w,

(3)

as devised by C. Fefferman (cf. [8]) by following the ideas of I. Naruki [9] as to associating to W the suspended domain

˜ = {(z , w, ζ ) ∈ W × C : |ζ |2(n+1) K (z , w, z , w) < 1}. W The pullback of G to W × S 1 , that is, j∗ G where j : W × S 1 ↪→ W × (C \{0}), is certainly degenerate. A fundamental question is then the following: Does j∗ G approach a nondegenerate (perhaps Lorentzian) metric F on ∂ W × S 1 as (z , w ) → ∂ W ? The answer in [8], that F is actually a Lorentzian metric on ∂ Ω × S 1 , is provided only for the case of a smoothly bounded strictly pseudoconvex domain Ω ⊂ C2 and relies on the asymptotic expansion of the Bergman kernel [10]. For ϵ > 0, the approximating domains Wϵ = {(z , w ) ∈ C2 : ρ (z , w ) + ϵ < 0}

are strongly pseudoconvex, as we will easily observe, and Fefferman’s asymptotic expansion formula applies to the Bergman kernel Kϵ (z , w, ζ , ω) to give Kϵ (z , w, z , w ) = cWϵ |∇ρ (z , w )|2 · det Lϵ (z , w ) · |ρϵ (z , w )|−3 + Eϵ (z , w, z , w )

(4)

for some constant cWϵ > 0 and some function Eϵ (z , w, ζ , ω) such that Eϵ ∈ C (W ϵ × W ϵ \ ∆ϵ ) and ∞

⏐ ⏐ ⏐ ⏐ ⏐Eϵ (z , w, z , w)⏐ ≤ c ′ |ρϵ (z , w)|−3+ 12 · ⏐ log|ρϵ (z , w)|⏐. Wϵ ′ Here Lϵ is the restriction of ∂∂ρ to T1,0 (∂ Wϵ ) ⊗ T0,1 (∂ Wϵ ), ∆ϵ is the diagonal of ∂ Wϵ × ∂ Wϵ , and cW > 0 is another constant ϵ (depending on Wϵ ). As an elementary consequence of the expansion (4),

ϕϵ (z , w) = −Kϵ (z , w, z , w)−1/3 is a defining function for Wϵ , though solely of class C 2 , implying that Fϵ =

lim

(z ,w )→∂ Wϵ

j∗ϵ Gϵ

is a Lorentzian metric on ∂ Wϵ × S 1 , whose restricted conformal class is a biholomorphic invariant of Wϵ , cf. [8]. It is of great interest to obtain some analog to the expansion (4) in the case of smooth, bounded, weakly pseudoconvex domains, and in particular for the worm domain, corresponding to the case ϵ = 0. An alternative approach is provided in [8] as well and relies on the possibility of replacing u(z , w ) in (2)–(3) by the (eventual) solution to the Dirichlet problem for the complex Monge–Ampère equation,

{

J(u) = 1 in W u = 0 on ∂ W ,

(5)

where u J(u) = (−1) det uz uw

(

n

uz uzz uw z

uw uz w uww

) .

The solution by S.-Y. Cheng & S.-T. Yau [11] to the problem (5) is confined to the case of a strictly pseudoconvex domain. S.-Y. Li [12] studied the same problem in the weakly pseudoconvex case, yet his approach requires a defining function which is plurisubharmonic on the entire boundary of the given pseudoconvex domain and none exist for W , cf. Proposition 2.2 in [2], p. 486. The solutions in [11] and [12] were unknown at the time [8] was published yet, as observed in [8], only the

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two-jet at the boundary of a solution u(z , w ) to (5) is required in the construction of F and an approximate solution of the sort may be built by setting

{ u=

1+

1 4

[

(

1−J

ρ J(ρ )

)]}

ρ J(ρ )

(6)

so that u = 1 + O(ρ 2 ), provided that (6), and in general the approximation scheme in [8], make sense for Ω = W . An elementary calculation shows that J(ρϵ )(0,w0 ) = −

ϵ |w0 |

for any ϵ ≥ 0 and any |log |w0 |2 | ≤ µ. In particular J(ρ ) vanishes along the annulus A and the alternative choice (6) is not available. Thus, in the context of the worm domain W ⊂ C2 , one misses the tools for understanding the behavior of j∗ G in the limit as (z , w ) → ∂ W and the extrinsic approach (as outlined above, cf. [8]) to a Fefferman-like metric on ∂ W × S 1 is an open problem. In this paper we consider M = ∂ W \ A, which an open subset of ∂ W and turns out to be a strictly pseudoconvex CR manifold. Observing that 2

2

ρ (z , w) = η(log |w|2 ) − e−i log |w| z − ei log |w| z + zz , we see that

θ=

i 2

j∗ ∂ − ∂ ρ ,

(

)

j : ∂ W ↪→ C2 ,

(7)

is a pseudohermitian structure on ∂ W whose pointwise restriction to M is a positively oriented contact form on M. Fefferman’s question (cf. [8]) whether an intrinsic approach to a Lorentzian metric, perhaps living on M × S 1 , is available when M is not the boundary of a domain was settled by J.M. Lee (cf. [13]). π Consider the total space C (M) of the canonical circle bundle S 1 → C (M) → M, and indeed it is globally diffeomorphic 1 to M × S . In [13] Lee built a Lorentzian metric Fθ ∈ Lor[C (M)] in terms of pseudohermitian invariants of (M , θ ), which is conformal equivalent to the (extrinsic) Fefferman metric, when the latter is defined. The Lorentzian manifold (C (M), Fθ ) admits a natural time orientation, hence it is a space–time. We shall provide details of this construction in Section 3. Let Γ : [a, b) → C (M) be a timelike curve in (C (M), Fθ ), parametrized by proper time t. Following (cf. [14]), we say that Γ runs into a curvature singularity if (i) Γ has bounded acceleration, (ii) λ ◦ Γ˙ is unbounded for some curvature invariant λ. The physical interpretation is that Γ , thought of as the world line for some particle or moving observer, encounters unbounded curvature as t → b and hence cannot be extended to proper time t = b, not in the space–time C (M) or any of its smooth extensions. If γ : [a, b) → M is a C 1 curve, by a lift of γ one means a C 1 curve Γ : [a, b) → C (M) such that π (Γ (t)) = γ (t) for any a ≤ t < b. ⏐ ⏐ Let w0 ∈ C \ {0} such that ⏐log |w0 |2 ⏐ ≤ µ. Let us set

[

I(w0 ) = log |w0 |2 , log |w0 |2 +

π) 2

⊂R

and let us consider

( [ ] ) γw0 (ϕ ) = 2 cos ϕ − log |w0 |2 eiϕ , w0 ,

ϕ ∈ I(w0 ),

which is a circle in M such as limϕ→log |w0 |2 +π /2 γw0 (ϕ ) does not exist in M — yet the limit is (0, w0 ) ∈ A in the topology of

∂W.

Finally, let D be the Levi-Civita connection of the Lorentzian manifold (C (M), Fθ ) and K its scalar curvature. Our first main result is the following. Theorem 1. Every lift Γ (ϕ ) ∈ C (M) of γw0 (ϕ ) ∈ M runs into a curvature singularity. Precisely, lim

ϕ→log |w0 |2 +π /2

K (Γ (ϕ )) = −∞,

(8)

i.e. Γ (ϕ ) encounters unbounded scalar curvature as ϕ → log |w0 |2 + π/2. Theorem 1 is proved in Section 3. Notice that Theorem 1 holds for an arbitrary lift of the circle γw0 . We may nevertheless show that γw0 may be lifted to a timelike curve Γ in (C (M), Fθ ), thus relating Theorem 1 to the general relativistic notions in [14].

E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

145

Theorem 2. There exist C ∞ curves Γ : I(w0 ) → C (M) such that: (i) π (Γ (ϕ )) = γw0 (ϕ ) for any ϕ( ∈ I(w0 ); ) (ii) Γ is timelike, that is, Fθ ,Γ (ϕ ) Γ˙ (ϕ ) (, Γ˙ (ϕ ) < 0 for) every ϕ ; (iii) Γ is future directed, that is, Fθ ,Γ (ϕ ) Γ˙ (ϕ ) , Xθ , Γ (ϕ ) < 0 for every ϕ , where Xθ is a time orientation of (C (M), Fθ ). The construction of the canonical time orientation Xθ of (C (M), Fθ ) is also deferred to Section 3. Theorem 2 is proved in Section 4. According to J.A. Thorpe points where a timelike or null curve Γ of bounded acceleration runs into a curvature singularity are b-boundary points. No proof of the statement is provided in [14]. Nevertheless, in view of Theorem 1, one may expect that limϕ→log |w0 |2 +π /2 Γ (ϕ ) exists in the topology of the b-completion C (M) and lies on the b-boundary C˙ (M) in the sense of G.B. Schmidt, [15]. The opposite scenario is however suggested by the following Theorem 3. The circle γw0 : I(w0 ) → M has infinite energy

( ) E γw0 =

log |w0 |2 + π2

∫

log |w0

|2

gθ , γ (ϕ ) γw ˙ 0 (ϕ ) , γw˙ 0 (ϕ ) dϕ = +∞

(

)

with respect to the Webster metric gθ on M. Therefore an observer attached to γw0 requires infinite energy to reach (0, w0 ) ∈ A. Theorem 3 is proved in Section 4. From now on, we write M = (C (M), Fθ ) for short, and let O(1, 3) → O(M) → M be the principal bundle of all Lorentzian frames. Let O++ (1, 3) and O+ (M) be connected components of O(1, 3) and O(M) respectively. Following [15], we let dG : O+ (M) × O+ (M) → [0, +∞) denote the Schmidt distance function and O+ (M) the ˙ of M is given by Cauchy completion of O+ (M) with respect to dG . The b-boundary M

]

[

˙ = O+ (M)/O++ (1, 3) \ M. M By a result in [15], b-boundary points may be characterized as end points of inextensible curves Γ : [a, b) → M, that is curves Γ for which limt →b− Γ (t) does not exist in the topology of M, whose horizontal lifts Γ H : [a, b) → O+ (M) have finite length b

∫

GΓ H (t) Γ˙ H (t) , Γ˙ H (t)

(

a

)1/2

dt < ∞

(9)

with respect to the Schmidt metric G (cf. [15]), which is a Riemannian metric on O+ (M) determined by the Levi-Civita connection 1-form of the Fefferman metric Fθ . Let Z = −ρw ∂/∂ z + ρz ∂/∂w be a generator of the CR structure T1,0 (∂ W ) and let Z = X − iY be its real and imaginary parts. Also let us set S = [(n + 2)/2] ∂/∂ s

(10)

with n = 1 (the convention in [16], p. 083504-19). Let Σ : M → O (M) be the cross section given by +

Σ (p)eα = Eα (p) ∈ Tp{(M), p ∈ M, } {Eα : 0 ≤ α ≤ 3} = T ↑ − S , E1↑ , E2↑ , T ↑ + S , √ √ 2

E1 =

g11

X,

E2 =

2

g11

(11)

Y.

↑

↑

Indeed, T ↑ − S is timelike while E1 , E2 and T ↑ + S are spacelike hence {Eα : 0 ≤ α ≤ 3} is a field of Lorentzian frames. Also, T ↑ and S are null. We are unable to prove or disprove (9) when Γ is a lift of γ = γw0 . However, the natural lift C = Σ ◦ Γ of any such Γ may be shown to have infinite length. Theorem 4. Let w0 ∈ C \ {0} such that ⏐log |w0 |2 ⏐ ≤ µ and let Γ : I(w0 ) → M be an arbitrary lift of the circle γw0 . If C : I(w0 ) → O+ (M) is defined by C = Σ ◦ Γ then

⏐

∫

log |w0 |2 + π2 log |w0

|2

GC (ϕ ) C˙ (ϕ ) , C˙ (ϕ )

(

)1/2

⏐

dϕ = +∞

where G is the Schmidt metric on O+ (M). Theorem 4 is proved in Section 5. The question whether Π −1 (A) contains b-boundary points of Fefferman’s space–time ˙ ̸= ∅] remains open. [i.e. whether Π −1 (A) ∩ M

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E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

So far we have exhibited (cf. Theorem 1) only curvature singularities of the Fefferman space–time M. Section 6 demonstrates a dimension reduction argument (similar in spirit to the approach in [17] and [18]) leading to the identification of subsets of the adapted bundle boundary ∂adt M. Theorem 5. Let Σ ⊂ M be a Lorentzian surface with a flat normal connection (R⊥ = 0). Let O(1, 1) → O(Σ ) → Σ and O(1, 1) × O(2) → O(M, Σ ) → Σ be respectively the principal bundle of Lorentzian frames tangent to Σ and the principal bundle of adapted Lorentzian frames tangent to M. Given a globally defined cross section sT : Σ → O(Σ ) there is a natural isometric immersion

j : O(Σ ) → O(M, Σ ),

j∗ Gadt = GΣ ,

(12)

with respect to the Schmidt metrics GΣ and Gadt . j descends to a map of orbit spaces

j:Σ =

O+ (Σ ) O++ (1, 1)

→

O+ (M, Σ ) O++ (1, 1)

× O+ (2)

=M

adt

(13)

such that

˙ ) ⊂ ∂adt M j(Σ

(14)

˙ = Σ \ Σ and ∂adt M = M adt \ M. where Σ The construction of the Schmidt metric Gadt on O(M, Σ ) and of the corresponding distance function dadt on O+ (M, Σ ) relies on the connection 1-form ω obtained as the o(1, 1) ⊕ o(2) component of i∗ j∗ ψ where ψ is the Levi-Civita connection 1-form of the Fefferman space–time M while j

i

O(M, Σ ) −→ O(M)|Σ −→ O(M) are injections. This is because i∗ j∗ ψ is not a connection 1-form on O(M, Σ ) and in particular the resulting bundle completions ˙ and ∂adt M) are logically unrelated. Theorem 5 is proved in Section 6. and bundle boundaries (i.e. M Section 7 is devoted to the geometry of the total space Σ of a principal circle subbundle S 1 → Σ → Γw , in S 1 → M → M, over the circle

Γw = {γw (ϕ ) : ϕ ∈ I(w ) \ {log |w|2 }},

⏐ ⏐ ⏐log |w|2 ⏐ < µ.

Such real surfaces Σ are shown to be examples to which our Theorem 5 applies. Theorem 6. For every |f (w )| < µ the immersion

ι : π −1 (Γw ) ↪→ M = (C (M), Fθ ) has a Lorentzian first fundamental form and a flat normal connection. Let sT : Σ → O(Σ ) be the section defined by sT (p)e0 = (dγ ↑ (ϕ ) Rζ )γ˙w↑ (ϕ ), w

p = γw↑ (ϕ ; p0 · ζ ) ∈ Σ ,

sT (p)e1 = Sp ,

p0 ∈ Σ ,

ζ ∈ S1 ,

2 4 where {e0( , e1 } ⊂ R orthonormal frame in the normal )⊥×{0} ⊂ R is⊥ the canonical linear basis. Let {N1 , N2 } be a globally defined −1 bundle T π (Γw ) such that ∇ Ni = 0 for i ∈ {1, 2} and let us consider the section s⊥ : Σ → O(T (Σ )⊥ ) defined by

s⊥ (p) : R2 → T π −1 (Γw )

(

s (p)ei+1 = Ni (p),

)⊥ p

p∈π

⊥

,

−1

(Γw ), i ∈ {1, 2},

where {e2 , e3 } ⊂ {0} × R ⊂ R . The sections (sT , s⊥ ) determine an isometric immersion 2

4

j : O π −1 (Γw ) → O M, π −1 (Γw )

(

)

(

)

descending to a map of bundle completions

j : π −1 (Γw ) → M

adt

which maps the b-boundary points of (π −1 (Γw ), , ι∗ Fθ ) into the adapted bundle boundary ∂adt M. The property j∗ Gadt = GΣ , where GΣ and Gadt are the Schmidt metrics (cf. [15]) on O(Σ ) and O(M, Σ ), claimed in Theorem 6 is essentially a consequence of

( ⊥ )∗ s

ψ⊥ = 0

(15)

where ψ is the normal connection 1-form of the immersion Σ ↪→ M. In turn (15) follows from the fact that the frame {N1 , N2 } determining s⊥ is parallel in the normal bundle. Such a frame may be chosen provided that Σ ↪→ M has a flat normal connection, which is the case for Σ = π −1 (Γw ) according to our Lemma 14. ⊥

E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

147

2. CR and pseudohermitian geometry We shall prove Theorems 1 and 2 by using techniques in pseudohermitian geometry (cf. e.g. [19]). We start by describing the CR and pseudohermitian structures on ∂ W and M. The CR structure T1,0 (∂ W )(z ,w) = T(z ,w) (∂ W )⊗R C ∩ T 1,0 (C2 )(z ,w) ,

[

(z , w ) ∈ ∂ W ,

]

2

induced by the complex structure on C , is globally the span of Z = −ρw ∂/∂ z + ρz ∂/∂w

(16)

where ρ is given by (1) and ρz = ∂ρ/∂ z etc. Let H(∂ W ) = Re T1,0 (∂ W ) ⊕ T0,1 (∂ W ) ,

{

T0,1 (∂ W ) ≡ T1,0 (∂ W ) ,

}

be the Levi, or maximally complex, distribution on ∂ W , thought of as endowed with the complex structure J : H(∂ W ) → H(∂ W ),

J V +V =i V −V ,

)

(

(

)

V ∈ T1,0 (∂ W ).

Let E(∂ W ) ⊂ T (∂ W ) be the conormal bundle associated to the Levi distribution i.e. the real line bundle given by ∗

E(∂ W )(z ,w) = ω ∈ T(z∗ ,w) (∂ W ) : Ker(ω) ⊃ H(∂ W )(z ,w)

{

}

for any (z , w ) ∈ ∂ W . Then θ , defined in (7), is a pseudohermitian structure on ∂ W i.e. a globally defined nowhere zero C ∞ section in E(∂ W ). The Levi form is Lθ V , W = −i (dθ ) V , W

(

)

(

)

for any V , W ∈ T1,0 (∂ W ). We also set Gθ (X , Y ) = (dθ )(X , JY ) for any X , Y ∈ H(∂ W ), so that Lθ is the same as the C-linear extension of Gθ to T1,0 (∂ W ) ⊗ T0,1 (∂ W ). Lemma 1. Let us set f (w ) = log |w|2 for any w ∈ C \ {0}. (i) The Levi invariant g11 = Lθ (Z , Z ) ∈ C ∞ (∂ W ) may be expressed as the third degree polynomial in z and z Fα1 α 2 (w ) z α1 z

∑

g11 =

α2

(17)

0≤|α|≤3

with Hermitian coefficients Fmn ∈ C ∞ (C, C) i.e. Fmn = Fnm given by F00 (w ) =

1

F20 (w ) = −

F21 (w ) =

[

2

F10 (w ) = −

F11 (w ) =

e−f (w) η′′ (f (w )) + η′ (f (w ))2 , 1 2 1 2

1 2 1

2 F30 (w ) = 0.

]

e−(1+i)f (w) η′′ (f (w )) − i η′ (f (w )) − 1 ,

[

]

e−(1+2i)f (w) ,

(18)

[ ] e−f (w) η′′ (f (w )) − 2 , e−(1+i)f (w) ,

(ii) If A = {(z , w ) ∈ ∂ W : (Lθ )(z ,w) = 0} is the null space of the Levi form, then A = {(0, w ) ∈ ∂ W :⏐ |f (w )| ≤ µ}. (iii) M = ∂ W \ A is a strictly pseudoconvex CR manifold, with the CR structure T1,0 (M) = T1,0 (∂ W )⏐M . (iv) Let Wϵ = {z , w ) ∈ C2 : ρ (z , w ) < −ϵ} with ϵ ≥ 0. There is ϵ0 > 0 such that {Wϵ : 0 ≤ ϵ ≤ ϵ0 } is a family of pseudoconvex domains with Wϵ strictly pseudoconvex for every 0 < ϵ ≤ ϵ0 . Proof. We first observe that the identity i {

(dθ )(Z , Z ) =

2

|ρw |2 ρzz + |ρz |2 ρww − ρw ρz ρz w − ρz ρw ρwz

}

together with

ρz = z − e−if (w) ,

ρzz = 1,

ρz w = i fw e−i f (w) ,

yields g11 =

i {( 2

+

1 − z e−i f (w) fw ρw − 1 − z ei f (w) fw ρw

)

1{ 2

(

)

}

} |ρw |2 + ρww (1 + |z |2 − z e−if (w) − z eif (w) ) .

(19)

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E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

Since

{ } ρw = fw η′ (f (w)) + i(z e−if (w) − z eif (w) ) , and

{ } ρww = |fw |2 η′′ (f (w)) + z e−i f (w) + z ei f (w) , Eq. (17) follows. Next for every ϵ ≥ 0 let ϕϵ be the local defining function for Wϵ given by

ϕϵ (z , w) = earg(w ) ρϵ (z , w) 2

with ρϵ = ρ + ϵ , so that ϕϵ = ϕ (1) + ϕ (2) + ϕϵ(3) where

ϕ (1) (z , w) = |z |2 earg(w ) , 2

ϕ (2) (z , w) = −2 Re(z e−i logw ), 2

ϕϵ(3) (z , w) = [η(f (w)) + ϵ ] earg(w ) . 2

The (locally well defined) function e− log w is holomorphic and its modulus is earg(w ) hence ϕ (1) and ϕ (2) are plurisubharmonic. Also ∆ϕϵ(3) ≥ 0 so ϕϵ(3) is plurisubharmonic, as well. So for ϵ0 > 0 sufficiently small each Wϵ , 0 ≤ ϵ ≤ ϵ0 , is a pseudoconvex domain, as claimed. Actually if ϵ > 0 then ϕϵ is strictly plurisubharmonic at every point (z , w ) because of ϕϵ(3) , while if ϵ = 0 then ϕ0 is strictly plurisubharmonic at each point (z , w) with z ̸= 0 (because of ϕ (1) and ϕ (2) ) or with |f (w)| > µ (because of ϕ0(3) ). Hence each Wϵ , 0 < ϵ ≤ ϵ0 is a strictly pseudoconvex domain and the weak pseudoconvexity locus of W is contained in 2

2

A = {(0, w ) : ⏐log |w|2 ⏐ ≤ µ} ⊂ ∂ W .

⏐

⏐

These are precisely the points where the Levi form Lθ vanishes, may be seen as a consequence of (17) and (18). This proves Lemma 1. □ By Lemma 1 the pullback of θ to M (denoted by the same symbol θ ) is a contact form on M i.e. θ ∧ dθ is a volume form on M. Let T ∈ X(M) be the Reeb vector field i.e. the globally defined, nowhere zero, vector field on M, transverse to the Levi distribution H(M), determined by

θ (T ) = 1, T ⌋ dθ = 0. Then T (M) = H(M) ⊕ RT . Let gθ be the Webster metric i.e. the Riemannian metric on M given by gθ (X , Y ) = Gθ (X , Y ),

gθ (X , T ) = 0,

gθ (T , T ) = 1,

for any X , Y ∈ H(M). Let ∇ be the Tanaka–Webster connection of (M , θ ) i.e. the unique linear connection on M satisfying: (i) the Levi distribution H(M) is parallel with respect to ∇ [i.e. Y ∈ H(M) H⇒ ∇X Y ∈ H(M) for any X ∈ X(M)]; (ii) ∇ T = 0, ∇ gθ = 0; (iii) the torsion tensor field T∇ is pure i.e. for any V , W ∈ T1,0 (M) and X ∈ X(M), T∇ (V , W ) = 0,

T∇ V , W = 2i Lθ V , W ,

(

)

(

)

τ ◦ J + J ◦ τ = 0,

where τ (X ) ≡ T∇ (T , X ) is the pseudohermitian torsion of (M , θ ). Let R∇ be the curvature tensor field of ∇ , defined as R∇ (X , Y ) = [∇X , ∇Y ] − ∇[X ,Y ] for any X , Y ∈ X(M). The pointwise restriction of Z [given by (16)] to M gives a frame T1 in T1,0 (M) and one sets A ∇ TB = ωB A ⊗ TA , ∇TB TC = ΓBC TA ,

where A, B, C , . . . ∈ 0, 1, 1 ,

{

}

T0 = T ,

T1 = Z ,

so that ωB and are respectively the connection 1-forms and Christoffel symbols of the Tanaka–Webster connection ∇ . As ∇ parallelizes both H(M) and J, it parallelizes the eigen-distributions T1,0 (M) and T0,1 (M) hence ω1 1 = 0 and ω1 1 = 0. Also ∇ T = 0 yields ω0 A = 0. Hence all 1-forms ωA B vanish except for ω1 1 and their complex conjugates ω1 1 = ω1 1 . Let θ 1 be the complex 1-form on M determined by A

A ΓBC

θ 1 (Z ) = 1,

θ 1 (Z ) = 0,

θ 1 (T ) = 0,

E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

149

and let us set θ 1 = θ 1 . Then {θ 1 , θ 1 , θ } is a global frame of T ∗ (M) ⊗ C and A θB , ωC A = ΓBC

θ 0 = θ.

As to the curvature components with respect to the frame {TA }, one sets R∇ T1 , T1 T1 = R1 1 11 T1 ,

(

)

R1111 = g11 R1 1 11 ,

Ric∇ (X , Y ) = Trace V ↦ → R∇ (V , Y )X ,

{

RAB = Ric∇ (TA , TB ),

}

R = g 11 R11 ,

X , Y ∈ X(M),

g 11 ≡ 1/g11 .

The terms R11 and R are respectively the pseudohermitian Ricci tensor and the pseudohermitian scalar curvature of (M , θ ). By a result of S.M. Webster (cf. [6]) τ maps T1,0 (M) into T0,1 (M) hence one may set

τ (T1 ) = A1 1 T1 ,

A11 = g11 A1 1 .

Lemma 2. The Christoffel symbols of the Tanaka–Webster connection of (M , θ ) are given by 1 = g 11 Z (g11 ), Γ11

1 Γ11 = 0,

(20)

1 1 2iρz g11 Γ01 = Γ11 (ρz ρz w − ρw )

( ) + ρz w (ρz ρz w − ρw ) − |ρz |2 ρz ww + 2 ρz ρww − |ρz w |2 .

(21)

Also Hα1 α 2 (w ) z α1 z

∑

Z (g11 ) =

α2

(22)

0≤|α|≤3

where Hmn ∈ C ∞ (C, C) are given by 1

η′ (f (w))F10 (w) − e−f (w) G00 (w), w ] 1 [ H10 (w ) = −2η′ (f (w))F20 (w) − ie−if (w) F10 (w) − −e−if (w) G10 (w), w ] 1 [ ′ −η (f (w))F11 (w) + ieif (w) F10 (w) + G00 (w) − e−if G01 (w), H01 (w ) = w H00 (w ) = −

H20 (w ) = −

2i

H21 (w ) = −

2i

e−if (w) F20 (w ) − e−if (w) G20 (w ), w ] 1 [ −ie−if (w) F11 (w) + 2ieif (w) F20 (w) − 2η′ (f (w))F21 (w) H11 (w ) = w + G10 (w ) − e−if (w) G11 (w ), ] 1 [ ′ −η (f (w))F12 (w) + ieif (w) F11 (w) + G01 (w) − e−if (w) G02 (w), H02 (w ) = w H30 (w ) = 0,

H12 (w ) = −

i

w

e−if (w) F21 (w ) + G20 (w ) − e−if (w) G21 (w ), e−if (w) F12 (w ) + G11 (w ) − e−if (w) G12 (w ),

eif (w) F12 (w ) + G02 (w ), w = ∂ Fmn /∂w for any 0 ≤ m + n ≤ 3.

H03 (w ) = and Gmn

i

w

Proof. Let us consider the functions 1 ρw − ρz ρz w α= , β = H α, H = . ρz − H ρw ρw ρz w − ρz ρww The Reeb vector field of (M , θ ) is T = i α ∂z − α ∂z − β ∂w + β ∂w .

{

}

(23)

By the purity axiom (in the description of ∇ )

[ ] 1 1 Γ11 Z − Γ11 Z − 2ig11 T = Z , Z .

(24)

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E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

On one hand Z , Z = (ρz ρww − ρw ρz w ) ∂z + (ρw ρzz − ρz ρz w ) ∂w

[

]

(25)

+ (ρw ρz w − ρz ρww ) ∂z + (ρz ρz w − ρw ρzz ) ∂w and on the other, by (23),

( 1 ) ( 1 ) 1 1 Z − Γ11 Z − 2ig11 T = Γ11 ρw + 2 α g11 ∂z − Γ11 ρz + 2 β g11 ∂w Γ11 ( ( ) ) 1 1 − Γ11 ρw + 2 α g11 ∂z + Γ11 ρz + 2 β g11 ∂w .

(26)

Substitution from (25)–(26) into (24) leads to 1 ρz ρww − ρw ρz w = Γ11 ρw + 2 α g11 ,

(27)

1 ρw ρzz − ρz ρz w = −Γ11 ρz − 2 β g11 .

(28)

Summing up (27)–(28) furnishes g11 =

1{

|ρz |2 ρww + |ρw |2 − ρz ρw ρz w − ρz ρw ρz w

2

}

(29)

which is easily seen to agree with (19). Next we substitute from (29) into (27) and obtain Γ 1 = 0, which is the second 11 identity in (20). In particular (again by (24)) Z , Z = −2i g11 T

[

]

(30)

which is often taken as a definition for the Levi invariant g11 . By a result in [19]

{ } 1 Γ11 = g 11 Z (g11 ) − gθ (Z , [Z , Z ]) which together with (30) yields the first identity in (20). To prove (22) one starts from

∂ g11 = F10 (w) + 2F20 (w)z + F11 (w)z + 2F21 (w)zz + F12 (w)z 2 , ∂z ∑ ∂ g11 α = Gα1 α 2 (w ) z α1 z 2 , ∂w

(31)

0≤|α|≤3

and exploits (18) to obtain G00 (w ) =

1 2w

G10 (w ) = −

G01 (w ) = −

G20 (w ) = G11 (w ) =

e−f (w) η′′′ (f (w )) − η′′ (f (w )) + 2η′′ (f (w ))η′ (f (w ))

{

} − η′ (f (w))2 ,

1 2w 1 2w

1 2w 1 2w 1

e−(1+i)f (w) η′′′ (f (w )) − (1 + 2i)η′′ (f (w ))

{

} − (1 − i)η′ (f (w)) + 1 + i , e−(1−i)f (w) η′′′ (f (w )) − (1 − 2i)η′′ (f (w ))

{

} − (1 + i)η′ (f (w)) + 1 − i ,

(1 + 2i)e−(1+2i)f (w) , e−f (w) η′′′ (f (w )) − η′′ (f (w )) + 2 ,

{

}

(1 − 2i)e−(1−2i)f (w) , 2w 1 G21 (w ) = − (1 + i)e−(1+i)f (w) , 2w 1 G12 (w ) = − (1 − i)e−(1−i)f (w) , 2w G30 (w ) = 0, G03 (w ) = 0.

G02 (w ) =

We are left with the proof of (21). By the very definition of pseudohermitian torsion, 1 [T , Z ] = Γ01 Z − A1 1 Z .

(32)

E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

151

On the other hand, by (23),

{ } [T , Z ] = i α ρz w − β ρww − α ρz w + β ρww + αz ρw − αw ρz ∂z { } + i α ρzz − β ρz w − α ρzz + β ρz w − βz ρw + βw ρz ∂w } } { { ∂α ∂β ∂β ∂α + ρz ∂z + i ρw − ρz ∂w + i −ρw ∂z ∂w ∂z ∂w

(33)

and 1 1 Z − A1 1 Z = Γ01 Γ01 (−ρw ∂z + ρz ∂w ) − A1 1 (−ρw ∂z + ρz ∂w ) .

(34)

Substituting (33) and (34) into (32), using the facts that ρzz = 0 and ρzz = 1, gives

{ } 1 Γ01 ρz = i −β ρz w − α + β ρz w + Z (β ) ,

(35)

and

} ∂β ∂β A1 ρz = i −ρw + ρz . ∂z ∂w {

1

Formulas (27), (28) with Γ 1 = 0 become 11

α=

1 2

g

11

{ρz ρww − ρw ρz w } ,

β=

1 2

g 11 {ρz ρz w − ρw ρzz } .

(36)

Moreover, by (36) and (20) it follows Z (β ) = −

+

1 2 1

1 g 11 Γ11 (ρz ρz w − ρw )

2

(37)

g 11 |ρz |2 ρz ww + ρz |ρz w |2 − ρww

{

(

)}

.

Finally, substituting (36) and (38) into (35) yields (21). □ 3. Pseudohermitian scalar curvature The curvature R1 1 11 and pseudohermitian Ricci curvature R11 of (M , θ ) are given by (cf. [6] or [19]) 1 1 1 1 1 1 1 ) + Γ11 ) − Z (Γ11 Γ11 − Γ11 Γ11 + 2i g11 Γ01 R11 = R1 1 11 = Z (Γ11 .

(38)

Consequently (by (20)) 1 1 ) + 2i g11 Γ01 R1 1 11 = −Z (Γ11 .

(39)

Let w0 ∈ C \ {0} such that log |w0 |2 ∈ Iµ . Let D(z0 ) = {z ∈ C : |z − z0 | < 1} be the unit disc of center z0 ∈ C and let Cw0 ⊂ ∂ W be the circle

[

(

Cw0 = ∂ D ei log |w0 |

=

{(

2

)]

× {w0 } [

reiϕ , w0 : r = 2 cos log |w0 |2 − ϕ , ⏐ϕ − log |w0 |2 ⏐ ≤

)

] ⏐

⏐

π} 2

.

Then Cw0 ∩ A = {(0, w0 )}. The key result, leading to (8) in Theorem 1, is the following. Proposition 1. Let w0 ∈ C \ {0} such that |f (w0 )| ≤ µ. Then R11 (reiϕ , w0 ) = −

2

|w0 |

+ O(r),

r → 0.

(40)

r → 0.

(41)

Consequently R(r eiϕ , w0 ) = −

4 r2

+ O(r −1 ),

Proof. The function η vanishes at f (w0 ) = log |w0 |2 to infinite order. Lemma 1 then gives 2 g11 (z , w0 ) = e−(1+i)f (w0 ) z + e−(1−i)f (w0 ) z − e−(1+2i)f (w0 ) z 2

− e−f (w0 ) zz − e−(1−2i)f (w0 ) z 2 + e−(1+i)f (w0 ) z 2 z + e−(1−i)f (w0 ) zz 2

152

E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

for any z ∈ ∂ D(ei f (w0 ) ). The above equation in polar coordinates z = rei ϕ is 1

g11 (z , w0 ) =

2

r 2 e−f (w0 ) .

(42)

To compute Z (g11 )(z ,w0 ) one uses (22) and the equations F00 (w0 ) = 0,

F11 (w0 ) = −e−f (w0 ) , G00 (w0 ) = 0, G01 (w0 ) = − G11 (w0 ) =

1

F10 (w0 ) =

2

F21 (w0 ) =

G10 (w0 ) = − 1−i

1

1+i 2 w0

e−(1−i)f (w0 ) ,

2w0

w0

e−(1+i)f (w0 ) ,

e−f (w0 ) ,

2

e−(1+i)f (w0 ) ,

G02 (w0 ) =

1+i

1 2

e−(1+2i)f (w0 ) ,

F30 (w0 ) = 0,

e−(1+i)f (w0 ) , 1 + 2i

G20 (w0 ) =

e−(1+i)f (w0 ) , 2w0 G30 (w0 ) = 0, G03 (w0 ) = 0, G21 (w0 ) = −

1

F20 (w0 ) = −

1 − 2i 2w0

2w0

e−(1+2i)f (w0 ) ,

e−(1−2i)f (w0 ) ,

G12 (w0 ) = −

1−i 2w0

e−(1−i)f (w0 ) ,

to derive H00 (w0 ) = 0, H01 (w0 ) =

1 2w0 1

H20 (w0 ) = −

e−f (w0 ) ,

2w0

H30 (w0 ) = 0, H12 (w0 ) =

1

H10 (w0 ) =

H11 (w0 ) = −

e−(1+3i)f (w0 ) ,

2w0

2w0

e−f (w0 ) ,

3+i 2w0

e−(1+i)f (w0 ) ,

H02 (w0 ) = −

2+i

H21 (w0 ) =

3 − 2i

e−(1+2i)f (w0 ) ,

2w0

2−i 2w0

e−(1−i)f (w0 ) ,

e−(1+2i)f (w0 ) ,

H03 (w0 ) =

1−i 2w0

e−(1−2i)f (w0 ) .

Therefore, Z (g11 )(z ,w0 ) =

1 − 2i 2w0

e−(1+i)f (w0 ) r 2 + O(r 3 ).

(43)

By (42), (43) and (20) we then obtain 1 Γ11 (z , w0 ) =

1 − 2i

w0

e−if (w0 ) + O(|z |)

for any z ∈ ∂ D(eif (w0 ) ). Next, note that 1 Γ11 (z , w0 )(ρz ρz w − ρw )(z ,w0 ) = −(2 + i) e−(1+i)f (w0 ) + O(r),

(44)

[ ] ρz w (ρz ρz w − ρw ) − |ρz |2 ρz ww + 2ρz (ρww − |ρz w |2 ) (z ,w

(45)

and 0)

−(1+i)f (w0 )

= 2e

+ O(r),

and substitution from (44)–(46) and ρz (z , w0 ) = re−iϕ − e−if (w0 ) into (21) gives 1 2i re−iϕ − e−if (w0 ) g11 (z , w0 )Γ01 (z , w0 ) = −i e−(1+i)f (w0 ) + O(r).

{

}

Let us multiply by ρz (γ (ϕ )) = reiϕ − ei f (w0 ) and take into account that |ρz (γ (ϕ ))| = 1. Then 1 g11 (z , w0 ) Γ01 (z , w0 ) =

1 2|w0 |2

+ O(r)

(46)

for any z = reiϕ ∈ D(eif (w0 ) ). In order to compute the curvature from (39), we use

{ (

⏐2 }

1 Z (Γ11 ) = (g 11 )2 Z Z (g11 ) g11 − ⏐Z (g11 )⏐

We need the following

)

⏐

.

(47)

E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

153

Lemma 3. For any z ∈ D(eif (w0 ) ) we have that

(

)

Z Z (g11 )

(z ,w0 )

7+i

=

2|w0 |4

r 2 + O(r 3 ).

(48)

Proof. We observe that

(

Kα1 α 2 (w )z α1 z

∑

)

Z Z (g11 ) =

α2

(49)

0≤|α|≤4

where, simply computations give that K00 (w ) = −

1

w

η′ (f (w))H01 (w ) − eif (w)

∂ H00 , ∂w

∂ H00 ∂H − eif (w) 10 , ∂w ∂w 2 ′ i if (w) if (w ) ∂ H01 H01 (w ) − e , K01 (w ) = − η (f (w ))H02 (w ) + e w w ∂w 1 i ∂ H10 ∂H K20 (w ) = − η′ (f (w ))H21 (w ) − e−if (w) H11 (w ) + − eif (w) 20 , w w ∂w ∂w K10 (w ) = −

K11 (w ) = −

1

w

2

w

η′ (f (w))H11 (w ) −

η′ (f (w))H12 (w ) −

i

e−if (w) H01 (w ) +

w

2i

w

e−if (w) H02 (w ) +

i

w

eif (w) H11 (w )

∂ H01 ∂H − eif (w) 11 , ∂w ∂w 3 ′ 2i if (w) if (w ) ∂ H02 H02 (w ) − e , K02 (w ) = − η (f (w ))H03 (w ) + e w w ∂w i ∂ H20 ∂H K30 (w ) = − e−if (w) H21 (w ) + − eif (w) 30 , w ∂w ∂w 2i −if (w) i if (w) ∂ H11 ∂H K21 (w ) = − e H12 (w ) + e H21 (w ) + − eif (w) 21 , w w ∂w ∂w 3i −if (w) 2i if (w) ∂ H02 if (w ) ∂ H12 K12 (w ) = − e H03 (w ) + e H12 (w ) + −e , w w ∂w ∂w 3i ∂ H03 K03 (w ) = eif (w) H03 (w ) − eif (w) . w ∂w +

In order to check (48), notice that by the first part of Lemma 3 K00 (w0 ) = 0, K01 (w0 ) =

1+i 2

e−(2−i)f (w0 ) ,

K11 (w0 ) = −

3i 2

e−2f (w0 ) ,

K10 (w0 ) =

1+i

2 3 + 2i

e−(2+i)f (w0 ) ,

e−2(1+i)f (w0 ) , 2 3 + i −(2−i)f (w0 ) K02 (w0 ) = − e . 2

K20 (w0 ) = −

Substitution into (49) then yields (48). Lemma 3 is proved. □ End of the Proof of Proposition 1. Notice that (43) implies

⏐ ⏐ ⏐Z (g )(z ,w ) ⏐2 = 5 e−3f (w0 ) r 4 + O(r 5 ). 11 0 4

(50)

Next let us substitute into (47) from (42) and (48)–(50) so that to derive 1 Z (Γ11 )(z ,w0 ) =

2+i + O(r). |w0 |2

(51)

Finally (39), (46) and (51) yield R11 (z , w0 ) = −

2

|w0 |2

+ O(r),

(52)

which is (40). Hence (by (42)) R(z , w0 ) = −

4 r2

+ O(r −1 ),

which is (41). Proposition 1 is therefore proved. □

(53)

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E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

A complex valued differential p-form ω on a CR manifold (N , T1,0 (N)) has type (p, 0) if T0,1 (N) ⌋ ω = 0. Let Λp,0 (N) → N be the bundle of all (p, 0)-forms. The canonical bundle K (N) → N is the complex line bundle K (N) = Λn+1,0 (N), the top degree (p, 0)-forms, where n is the CR dimension of N. Next R+ acts on K (N) so that the quotient space C (N) = [K (N) \ {zero section}] /R+ is the total space of a S 1 -bundle over N, called the canonical circle bundle, with projection ΠN : C (N) → N. Let W ⊂ C2 be a worm domain and let C (∂ W ) be the canonical S 1 -bundle over ∂ W , with projection Π = Π∂ W . C (∂ W ) carries a natural foliation V by circles, called the vertical foliation, tangent to the distribution Ker(dΠ ). We also recall that a subset of a foliated manifold is called saturated if it is a union of leaves. Lemma 4. It holds that C (M) = Π −1 (M). In particular C (M) is an open saturated subset of (C (∂ W ), V ). Recall that we set M = ∂ W \ A. By construction, the metric Fθ is (cf. Definition 2.15 in [19], p. 128)

˜ θ + 2 (π ∗ θ ) ⊙ σ , Fθ = π ∗ G σ =

1{

d s + π ∗ σ0 ,

3

}

(54)

σ0 = i ω1 1 −

i 2

g 11 dg11 −

R 8

θ,

(55)

where s is a local fiber coordinate on C (M). Also ω1 1 , g11 and R are respectively the connection 1-forms of the Tanaka–Webster connection of the pseudohermitian manifold (M , θ ), the Levi invariant, and the pseudohermitian scalar curvature. Let M be a 4-dimensional manifold. A Lorentzian metric on M is a nondegenerate symmetric smooth (0, 2)-tensor field F on M of signature (− + + +). A tangent vector v ∈ Tp (M) is spacelike, resp. timelike, resp. null, if Fp (v, v ) > 0, resp. if Fp (v, v ) < 0, resp. v ̸ = 0 and Fp (v, v ) = 0. A vector field X , defined on some open subset U ⊂ M is called timelike if Xp is a timelike tangent vector for each p ∈ U . A globally defined timelike vector field X on M is referred to as a time orientation of the Lorentzian manifold (M, F ) and the synthetic object (M, F , X ) is a space–time. A tangent vector v ∈ Tp (M) is future, resp. past, directed if Fp (v, Xp ) < 0, resp. if Fp[(v,( Xp ) >)0. ] The pointwise restriction of θ = ι∗ 2i ∂ − ∂ ρ to M is a positively oriented contact form on M. C.R. Graham recognized (cf. [20]) σ as a connection 1-form in the principal bundle S 1 → C (M) → M. Let X ↑ ∈ X(C (M)) denote the horizontal lift of X ∈ X(M) with respect to σ i.e.

σ (X ↑ ) = 0, (dp π )Xp↑ = Xπ (p) , p ∈ C (M). If S ∈ X(C (M)) is the tangent to the S 1 action and T ∈ X(M) the Reeb vector field of (M , θ ) then Xθ ≡ T ↑ − S is a time orientation of (C (M), Fθ ), hence (C (M), Fθ , Xθ ) is a space–time. Let K be the scalar curvature of Fθ . As S 1 ⊂ Isom(C (M), Fθ ) it follows that K is S 1 -invariant. Hence, there is a unique function π∗ K ∈ C ∞ (C (M), R) whose vertical lift is K . By a result of J.M. Lee, [13] (cf. also [19], p. 142)

π∗ K = 3R/2.

(56)

Finally for any lift Γ of γw0 lim

ϕ→f (w0 )+π /2

K (Γ (ϕ )) =

3 2

lim

ϕ→f (w0 )+π /2

R(γw0 (ϕ )) = −∞

as a consequence of Proposition 1. Theorem 1 is proved. 4. Existence of timelike lifts To start with one produces a local coordinate neighborhood (U , xj ) on M such that γw0 (ϕ ) ∈ U for any 0 ≤ ϕ−f (w0 ) < π/2 i.e. Lemma 5. The set U =

{(

r ei ϕ (r ,w) , w : r > 0, w ∈ C∗ , r 2 − 2r + η(f (w )) < 0 ,

)

ϕ (r , w) ≡ f (w) + arccos

}

[

r 2

+

η(f (w)) 2r

]

,

(57) (58)

is an open subset of M and the map

χ : U → (0, +∞) × C∗ ,

χ (z , w) = (|z | , w ), (z , w) ∈ U ,

(59)

is a homeomorphism on χ (U) with the inverse

( ) χ −1 (r , w) = r ei ϕ (r ,w) , w ,

(r , w ) ∈ χ (U).

The curve γ = γw0 lies in U i.e. γ (ϕ ) ∈ U for any 0 ≤ ϕ − f (w0 ) <

π 2

.

E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

155

Proof. Let us set z = reiϕ . Then ρ (z , w ) = 0 yields cos [ϕ − f (w )] =

r

+

2

η(f (w)) 2r

where one may solve for ϕ provided that r 2 − 2r + η(f (w )) ≤ 0 so that ϕ = ϕ (r , w ) where ϕ (r , w ) is given by (58). The construction of the fiber coordinate s in (55) may be detailed as follows. For an arbitrary fixed s0 ∈ R

λ , |λ|

|s − s0 | < π, [ ( ) ] p = λ θ ∧ θ 1 x , x = (z , w ) ∈ U , ( λ ) Φ : C (M) → M × S 1 , Φ (p) = x , , |λ| U ≡ Φ −1 (M × V (s0 )) ⊂ M, V (s0 ) = {eis : |s − s0 | < π}. s : U → (s0 − π , s0 + π ) ,

s(p) = s,

eis =

The local coordinates corresponding to the local chart χ in Lemma 5 are denoted by

) ( χ = (r , w) = x˜ 1 , x˜ 2 + i x˜ 3 . Let xj = x˜ j ◦ π : U → R with U = Φ −1((U × V (0)) so)that (U , xj , s) is an induced local coordinate system on C (M). We often set x0 = s so that {xα : 0 ≤ α ≤ 3} ≡ s, x1 , x2 , x3 . Lemma 6. Let us set θ 1 = β dz + α dw where

α=

1

ρz − H ρw

,

β = Hα, H =

ρw − ρz ρz w . ρw ρz w − ρz ρww

Then θ 1 (Z ) = 1, θ 1 (Z ) = 0 and θ 1 (T ) = 0 so that {θ 1 , θ 1 , θ } is an admissible frame of T ∗ (M) ⊗ C. Here θ 1 = θ 1 . The proof is a straightforward calculation. Any lift Γ of γ = γw0 may be written as

[ ( ) ] Γ (ϕ ) = λ(ϕ ) θ ∧ θ 1 γ (ϕ ) for some C ∞ function λ : I(w0 ) → C. Let us set

Γ α (ϕ ) = xα (Γ (ϕ )),

0 ≤ α ≤ 3,

so that eiΓ

0 (ϕ )

=

λ(ϕ ) , Γ j (ϕ ) = γ j (ϕ ), |λ(ϕ )|

1 ≤ j ≤ 3,

where γ j = xj ◦ γ . Lemma 7. The tangent vector to γ = γw0 may be represented as

γ˙ (ϕ ) = 2 Tγ (ϕ ) −

1

1

2 cos2 [ϕ − f (w0 )]

{

w0 ei f (w0 ) Z + w0 e−i f (w0 ) Z

}

γ (ϕ )

.

(60)

Consequently the tangent vector to any lift Γ : I(w0 ) → C (M) of γ is given by

[

Γ˙ (ϕ ) = γ˙ (ϕ )↑ +

dΓ 0 dϕ

](

(ϕ ) + σ0, γ (ϕ ) (γ˙ (ϕ ))

∂ ∂s

) (61) Γ (ϕ )

where horizontal lifting is meant with respect to σ . The proof of Lemma 7 follows by a rather involved evaluation of the first and second order derivatives of ρ along the circle

{ } γ : z = eif (w0 ) 1 + e2i[ϕ−f (w0 )] ,

w = w0 ,

(62)

that is,

ρz (γ (ϕ )) = e−i[2ϕ−f (w0 )] , } i { 2i[ϕ−f (w0 )] ρw (γ (ϕ )) = e − e−2i[ϕ−f (w0 )] , w0 ρz w (γ (ϕ )) = ρww (γ (ϕ )) =

i

w0

e−i f (w0 ) ,

1

|w0 |

)2

ei[ϕ−f (w0 )] + e−i[ϕ−f (w0 )] .

( 2

(63)

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E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

Also if G = 1/H then β = 1/(G ρz − ρw ) hence, using (63), G(γ (ϕ )) = −

β (γ (ϕ )) =

i

eiϕ e2i[ϕ−f (w0 )] ei[ϕ−f (w0 )] + e−i[ϕ−f (w0 )]

{

w0

}

{ } × e2i[ϕ−f (w0 )] + e−2i[ϕ−f (w0 )] , e−2i[ϕ−f (w0 )]

iw0

(64)

.

4 cos2 [ϕ − f (w0 )]

The tangent vector along (62) is given by

} { γ˙ (ϕ ) = 2i ei[2ϕ−f (w0 )] (∂z )γ (ϕ ) − e−i[2ϕ−f (w0 )] (∂z )γ (ϕ ) .

(65)

Let us apply

θ=

i 2

(ρz dz − ρz dz + ρw dw − ρw dw)

(66)

to (65) and take into account the first equation in (63). We obtain

θγ (ϕ ) (γ˙ (ϕ )) = 2.

(67)

Similarly, one applies θ 1 = β dz + α dw to (65) so that to get

θγ1(ϕ ) (γ˙ (ϕ )) = 2i β (γ (ϕ )) ei[ϕ−f (w0 )] and then, using (64),

θγ1(ϕ ) (γ˙ (ϕ )) = −

ei f (w0 )

w0

2 cos2 [ϕ − f (w0 )]

.

(68)

Finally (67) and (68) yield (60). In order to prove (61) one starts from

Γ˙ (ϕ ) =

dΓ 0 dϕ

(ϕ )

(

∂ ∂s

) + Γ (ϕ )

dγ j dϕ

(ϕ )

(

∂ ∂ xj

) Γ (ϕ )

.

(69)

On the other hand, by taking into account the direct sum decomposition TΓ (ϕ ) (C (M)) = Ker(σ )Γ (ϕ ) ⊕ Ker dΓ (ϕ ) π ,

(

)

one may represent Γ˙ (ϕ ) as ↑ Γ˙ (ϕ ) = VΓ (ϕ ) + µ(ϕ ) SΓ (ϕ ) ,

(70)

for some tangent vector field V ∈ X(M) and some smooth function µ : I(w0 ) → R. Substitution from (69) into (70) followed by applying σΓ (ϕ ) yields Vγ (ϕ ) = γ˙ (ϕ ),

µ(ϕ ) =

dΓ 0 dϕ

(ϕ ) + σ0, γ (ϕ ) (γ˙ (ϕ ))

and Lemma 7 is proved. □ Theorem 3 is an immediate corollary to (60) in Lemma 7. Indeed gθ , γ (ϕ ) (γ˙ (ϕ ), γ˙ (ϕ )) = 4 +

1 cos2

[ϕ − f (w0 )]

hence E(γ ) = +∞. ˜ θ = Gθ Next, we need to compute the length of Γ˙ (ϕ ) with respect to the Fefferman metric Fθ . To this end we recall that G ˜ on H(M) ⊗ H(M) and G(T , V ) = 0 for any V ∈ X(M). Also, Gθ (Z , Z ) = 0,

Gθ (Z , Z ) = g11 ,

hence, by Lemma 7,

( ) π ∗ G˜ θ

Γ (ϕ )

( ) |w0 |2 Γ˙ (ϕ ) , Γ˙ (ϕ ) = 2

g11 (γ (ϕ )) cos4 [ϕ − f (w0 )]

.

(71)

Also (as previously shown)

( ∗ ) π θ Γ (ϕ ) Γ˙ (ϕ ) = 2,

(72)

E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

σΓ (ϕ ) Γ˙ (ϕ ) =

1

[

dΓ 0 dϕ

3

157

]

(ϕ ) + σ0,γ (ϕ ) (γ˙ (ϕ )) .

(73)

The formulae (71)–(73), together with (54) yield Fθ ,Γ (ϕ ) Γ˙ (ϕ ) Γ˙ (ϕ ) =

(

)

g11 (γ (ϕ ))

|w0 |2

[[ϕ −0f (w0 )] ] dΓ + (ϕ ) + σ0,γ (ϕ ) (γ˙ (ϕ )) . 3 dϕ cos4

2

(74)

4

As a byproduct of the proof of Proposition 1 we obtain g11 (z , w0 ) =

1 2

|z | e−f (w0 ) ,

hence g11 (γ (ϕ )) =

2

|w0 |2

cos2 [ϕ − f (w0 )] .

(75)

By (74) and (75), it follows that Γ is timelike if and only if 4

[

dΓ 0 dϕ

3

]

(ϕ ) + σ0,γ (ϕ ) (γ˙ (ϕ ))

cos2 [ϕ − f (w0 )] < −1.

(76)

Finally, by integrating in (76), if g : I(w0 ) → R is an arbitrary strictly decreasing smooth function,

Γ (ϕ ) = g(ϕ ) − 0

3 4

tan [ϕ − f (w0 )] −

ϕ

∫

f (w0 )

σ0,γ (t) (γ˙ (t)) dt

(77)

then Γ is timelike. For any lift Γ of γ given by (77), by (76), one has Fθ , Γ (ϕ )

(

] [ ) 1 dΓ 0 ˙ Γ (ϕ ) , Xθ , Γ (ϕ ) = −1 + (ϕ ) + σ0,γ (ϕ ) (γ˙ (ϕ )) < 0 3 dϕ

hence Γ is future directed. Theorem 2 is proved. 5. Levi flat points and space–time singularities ΠL

Let M = C (M) and let GL(4, R) → L(M) −→ M be the principal bundle of all linear frames tangent to M. An element u ∈ L(M) with ΠL (u) = p is an R-linear isomorphism u : R4 → Tp (M). Let ψ ∈ C ∞ (T ∗ (L(M)) ⊗ gl(4, R)) be the Levi-Civita connection 1-form of the Fefferman metric Fθ i.e.

( ) ψ A∗ = A,

R∗a ψ = ad(a−1 ) ψ,

for any left invariant vector field A ∈ gl(4, R) and any a ∈ GL(4, R). Let Θ ∈ C ∞ T ∗ (L(M)) ⊗ R4 be the canonical 1-form i.e.

(

Θu = u−1 ◦ (du ΠL ) ,

)

u ∈ L(M).

The Schmidt metric (cf. [15]) is the Riemannian metric G on L(M) given by G(V , W ) = ψ (V ) · ψ (W ) + Θ (V ) · Θ (W ) for any V , W ∈ T (L(M)) and the ‘‘dot-products’’ are the Euclidean inner products on R16 ≈ gl(4, R) and R4 . As M is oriented, L(M) has two connected components L± (M). Let GL+ (4) be the Lie group consisting of all a ∈ GL(4, R) such that det(a) > 0. Let dG : L+ (M) × L+ (M) → [0, +∞) be the distance function associated to the Riemannian metric G i.e. dG (u, v ) is the infimum of lengths of all piecewise C 1 curves joining u, v ∈ L+ (M). Let L+ (M) be the Cauchy completion of L+ (M) with respect to the distance function dG . For each a ∈ GL+ (4) the right translation Ra : L+ (M) → L+ (M) is uniformly continuous with respect to dG so that the action of GL+ (4) on L+ (M) extends to L+ (M) as a topological group action. Let

M = L+ (M)/GL+ (4) be the orbit space. There is a natural injection

M ≈ L+ (M)/GL+ (4) ↪→ L+ (M)/GL+ (4).

˙ = M \ M. A frame u ∈ L+ (M) with ΠL (u) = p is Lorentzian if The b-boundary of the space–time M is defined as M Fθ , p u(eα ) , u(eβ ) = ϵα δαβ ,

(

ϵ0 = −1 = −ϵj ,

)

1 ≤ j ≤ 3.

0 ≤ α , β ≤ 3,

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E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

Let O(1, 3) be the Lorentz group and O(M) → M the principal O(1, 3)-bundle of Lorentizan frames tangent to M. Let Σ : M → L(M) be the section given by (11). Note that Σ (M) ⊂ O(M) i.e. Σ (p) is a Lorentzian ∑n−1 i i frame for any p ∈ M. We adopt the notation Rn1 for Rn together with the Minkowski form η1,n−1 = −x0 y0 + i=1 x y of index ν = 1 [so that O(1, n − 1) consists of all a ∈ GL(n, R) preserving η1,n−1 ]. Each a ∈ O(1, 3) may be represented as

( a=

aT c

b aS

)

where aT : R11 → R11 (the timelike part of a) is the restriction of a to R11 composed with the orthogonal projection on R11 while aS : R3 → R3 (the spacelike part of a) is the restriction of a to R3 followed by orthogonal projection on R3 . We set as customary O++ (1, 3) = {a ∈ O(1, 3) : aT (1) > 0, det(aS ) > 0} , which is a connected component of O(1, 3). Next let us fix a connected component O+ (M) such that O+ (M) is a submanifold of L+ (M) and Σ (M) ⊂ O+ (M). Schmidt’s metric G induces a Riemannian metric on O+ (M) [the first fundamental form of ι : O+ (M) ↪→ L+ (M)] and hence a distance function dι∗ G . By a result in [15] the b-boundary of Fefferman’s space–time M may be recovered as

[

] [

˙ = O+ (M)/O++ (1, 3) \ O+ (M)/O++ (1, 3) M

]

where O+ (M) is the Cauchy completion of O+ (M) with respect to dι∗ G . Let Γ : I(w0 ) → M be an arbitrary lift of the circle γ = γw0 i.e. π ◦ Γ = γ . Let Γ H : I(w0 ) → O+ (M) be a horizontal lift of Γ with respect to the connection 1-form ω ∈ C ∞ (T ∗ (O+ (M)) ⊗ o(1, 3)) [the Levi-Civita connection 1-form of Fθ ] i.e. ΠO+ ◦ Γ H = Γ and Γ˙ H (ϕ ) is a horizontal vector. ( ) Here ΠO+ : O+ (M) → M is the projection. As ωΓ H (ϕ ) Γ˙ H (ϕ ) = 0 the length of Γ H is f (w0 )+π /2

∫

f (w0 )

(

f (w0 )+π /2

∫ =

f (w0 ) f (w0 )+π /2

∫

)1/2

GΓ H (ϕ ) Γ˙ H (ϕ ) , Γ˙ H (ϕ )

= f (w0 )

dϕ

 ( ) ΘΓ H (ϕ ) Γ˙ H (ϕ )  dϕ  H −1  Γ (ϕ ) Γ˙ (ϕ ) dϕ,

where ∥ξ ∥ is the Euclidean norm of ξ ∈ R4 . Together with a result in [15] this yields

˙ if and only Theorem 7. The limit limϕ→log |w0 |2 +π /2 Γ (ϕ ) exists in the M topology and determines a point on the b-boundary M if the improper integral π /2

∫

 H  Γ (t + log |w0 |2 )−1 Γ˙ (t + log |w0 |2 ) dt

0

is convergent. However one may exhibit another class of lifts of Γ to O+ (M) reaching their end point in infinite proper time. Precisely let C : I(w0 ) → O+ (M) be given by C = Σ ◦ Γ . Let us endow M, M and L+ (M) respectively with )the local coordinate ( systems (U , x˜ i ), (U , xµ ) [induced by (U , x˜ i ) i.e. xi = x˜ i ◦ π and x0 = s, cf. Section 4] and ΠL−1 (U , X µ , Xνµ [induced by (U , xµ ) i.e. X µ = xµ ◦ ΠL and Xνµ are the fiber coordinates]. Let C µ (ϕ ) = X µ (C (ϕ )) = Γ µ (ϕ ),

Cνµ (ϕ ) = Xνµ (C (ϕ )),

be the local components of C . If V is a tangent vector field on M then V H denotes its horizontal lift with respect to ω i.e. VuH ∈ Ker(ωu ),

(du ΠO+ )VuH = VΠO+ (u) ,

u ∈ O+ (M).

Also if v ∈ Tp (M) and u ∈ O+ (M) is a Lorentzian frame with origin at ΠO+ (u) = p then the horizontal lift of v is v H = βu (v ) where βu is the R-linear isomorphism

[ ]−1 βu ≡ du ΠO+ : Ker(ωu ) → Tp (M) . The horizontal lift of ∂/∂ xα is given by (cf. e.g. [21], vol. I)

∂ ∂ xα { }

(

)H =

∂ ∂Xα

} ∂ µ Xβ αβ ν ∂ Xνµ

{ −

(78)

µ

where αβ are the Christoffel symbols of gµν = Fθ (∂µ , ∂ν ) with ∂µ ≡ ∂/∂ xµ . Then, (78) implies

[ µ { } ]( ) α ∂ ˙C (ϕ ) = Γ˙ (ϕ )H + dCν (ϕ ) + dΓ (ϕ ) µ (Γ (ϕ ))Cνβ (ϕ ) . µ αβ dϕ dϕ ∂ Xν C (ϕ )

(79)

E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

159

Let us set

) ( ξ (ϕ ) = ΘC (ϕ ) C˙ (ϕ ) ∈ R4 . If ξ (ϕ ) = ξ α (ϕ )eα then

[ ] Γ˙ (ϕ ) = ξ α (ϕ ) Eα (Γ (ϕ )) = −ξ 0 (ϕ ) + ξ 3 (ϕ ) SΓ (ϕ ) +

{[

(80)

] }↑ ξ 0 (ϕ ) + ξ 3 (ϕ ) T + ξ 1 (ϕ ) E1 + ξ 2 (ϕ ) E2 Γ (ϕ ) .

Here S is given by (10) in Section 1. Also note that

ξ 1 (ϕ ) E1,γ (ϕ ) + ξ 2 (ϕ ) E2,γ (ϕ ) = √

1

[

2g11

ζ (ϕ ) Zγ (ϕ ) + ζ (ϕ ) Z γ (ϕ )

]

(81)

where ζ = ξ 1 + i ξ 2 . Then, by (61), (80) and (81), we have 2 3

[

dΓ 0 dϕ

] (ϕ ) + σ0,γ (ϕ ) (γ˙ (ϕ )) = −ξ 0 (ϕ ) + ξ 3 (ϕ ),

and

[ ] γ˙ (ϕ ) = ξ 0 (ϕ ) + ξ 3 (ϕ ) Tγ (ϕ ) + √

1

[

2g11 (γ (ϕ ))

] ζ (ϕ ) Zγ (ϕ ) + ζ (ϕ ) Z γ (ϕ ) ,

yielding, using (60) and (75),

ζ (ϕ ) = −

w0 eif (w0 ) , |w0 | cos [ϕ − f (w0 )]

(82)

and

ξ 3,0 (ϕ ) = 1 ±

1

[

dΓ 0 dϕ

3

(ϕ ) + σ0,γ (ϕ ) (γ˙ (ϕ ))

] (83)

by (82)–(83) we may conclude that

  ΘC (ϕ ) C˙ (ϕ )2 = 2 +

2 dΓ 0

[

1 cos2 [ϕ − f (w0 )]

+

9

dϕ

]2 (ϕ ) + σ0,γ (ϕ ) (γ˙ (ϕ )) .

Finally,

∫

f (w0 )+π /2 f (w0 )

GC (ϕ )

)1/2 C˙ (ϕ ), C˙ (ϕ ) dϕ ≥

(

∫

f (w0 )+π /2 f (w0 )

∫

f (w0 )+π /2

≥ f (w0 )

  ΘC (ϕ ) C˙ (ϕ ) dϕ dϕ cos [ϕ − f (w0 )]

= ∞. Theorem 4 is then proved. 6. Dimension reduction and b-boundary points Section 6 is devoted to a dimension reduction argument, leading to the determination of subsets of the adapted bundle boundary ∂adt M, that we will introduce shortly. The approach is inspired by the work of R.A. Johnson [17] and B. Bossard [18]. Most of the theoretical considerations in Section 6, such as induced and normal connections, bundle boundary constructions, etc., apply to any Lorentzian surface Σ in (M, Fθ ) such that the immersion Σ ↪→ M has a flat normal connection. Natural examples to keep in mind are saturated subsets of M got as vertical lifts of circles γw0 i.e. total spaces of principal circle subbundles S 1 → Σ → Γw0 . Precisely for each |f (w0 )| ≤ µ we set

{ } Γw0 = γw0 (ϕ ) : ϕ ∈ I(w0 ) \ {f (w0 )} . ( ) Then Σ = π −1 Γw0 ≈ (0 , π2 ) × S 1 is a real hypersurface in M.(It will be useful to represent Σ as the translation by the ) S 1 -action of the σ -horizontal lift of γ = γw0 . To this end, let x0 = 2eif (w0 ) , w0 ∈ M so that γ issues at x0 . Let p0 ∈ π −1 (x0 ) and let

γ ↑ = γ ↑ ( · ; p0 ) : I(w0 ) → M be the σ -horizontal lift of γ issuing at p0 .

160

E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

Lemma 8. (i) Set

} { Σ (p0 ) = γ ↑ (ϕ; p0 ) · ζ : ϕ ∈ I(w0 ) \ {f (w0 )}, ζ ∈ S 1 . Then Σ = Σ (p0 ). (ii) Σ is a Lorentzian surface with the induced metric ι∗ Fθ , where ι : Σ ↪→ M. In particular if T (Σ )⊥ → Σ is the normal bundle of ι then Fθ ,p is positive definite on T (Σ )⊥ p . (iii) For any point p = γ ↑ (ϕ ) · ζ ∈ Σ

{ |w0 | X↑ + cos [ϕ − f (w0 )] { |w0 | N2 (p) = Y↑ + cos [ϕ − f (w0 )]

N1 (p) =

1

[

1

e−if (w0 ) +

2 w0 [ 1 i 2

e−if (w0 ) −

w0

1

w0 1

w0

] }

eif (w0 ) S

] }p

eif (w0 ) S

, (84)

,

p

is an orthonormal basi in the normal space at p, that is, Fθ (Ni , Nj ) = δij for 1 ≤ i, j ≤ 2. Proof. (i) The proof is straightforward. Note also that given q0 ∈ π −1 (x0 ) there is ζ ∈ S 1 such that q0 = p0 · ζ and hence

γ ↑ (ϕ; q0 ) = γ ↑ (ϕ; p0 ) · ζ . (ii) For every 0 < ϕ − f (w0 ) < π2 the tangent vectors γ˙ ↑ (ϕ ) and Sγ ↑ (ϕ ) are linearly independent. Indeed γ˙ ↑ (ϕ ) ∈ Ker(σ )γ ↑ (ϕ ) and Sγ ↑ (ϕ ) ∈ Ker(dγ ↑ (ϕ ) π ) and Ker(σ ) ∩ Ker(dπ ) = (0). Let p = γ ↑ (ϕ ) · a ∈ Σ and V ∈ Tp (Σ ). Then V = λ γ˙ ↑ (ϕ ; p0 · a) + µ Sp

(85)

for some λ, µ ∈ R. Assume that Fθ ,p (V , W ) = 0 for any W ∈ Tp (Σ ). Then, 0 = Fθ ,p V , γ˙ ↑ (ϕ ; p0 · a)

(

)

= λ G˜ θ ,γ (ϕ ) (γ˙ (ϕ ) , γ˙ (ϕ )) + µ θγ (ϕ ) (γ˙ (ϕ )) σ (S)p λ + µ, = cos2 [ϕ − f (w0 )] and 0 = Fθ ,p (V , Sp ) = (λ/2) θγ (ϕ ) (γ˙ (ϕ )) = λ, yield λ = µ = 0. It follows that (ι∗ Fθ )p is nondegenerate for any p ∈ Σ . In order to see that the scalar product (ι∗ Fθ )p has index 1 it suffices to produce a timelike vector in Tp (Σ ). Indeed if V is given by (85) then

λ2 + 2λµ [ϕ − f (w0 )] hence Fθ ,p (V , V ) < 0 for suitably chosen λ and µ. Fθ ,p (V , V ) =

cos2

(iii) With the notations in Section 1 one may look for N ∈ C ∞ (T (Σ )⊥ ) as N = λa Ea↑ + µ+ T ↑ + S + µ− T ↑ − S

(

(

)

)

for some λa , µ± ∈ C ∞ (Σ , R). The tangent space Tp (Σ ) is the span of γ˙ ↑ (ϕ; p0 · ζ ), Sp . Then

{

Fθ ,p (N , S)p = 0,

µ+ + µ− = 0, √ µ+ − µ− =

Fθ ,p (Np , γ˙ ↑ (ϕ; p0 · ζ )) = 0,

g11

1

2 2 cos2 [ϕ − f (w0 )]

{

} w0 β eif (w0 ) + w0 β e−if (w0 ) ,

with β = λ1 − i λ2 . Therefore,

{ } 1 −i(f (w0 )) β |w0 | ↑ N = Z + e S 2 cos[ϕ − f (w0 )] w0 { } 1 i(f (w0 )) β |w0 | ↑ + Z + e S 2 cos[ϕ − f (w0 )] w0 and hence N = 2λa Na where Na are given by (84). □ Let t ∈ R and let

( Bt =

cosh t sinh t

sinh t cosh t

)

}

E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

161

be the boost of R21 through (oriented) Lorentz angle t. Then O++ (1, 1) = {Bt : t ∈ R} is the component of the identity in the Lorentz group O(1, 1). Let ι : Σ ↪→ M be a Lorentzian surface and let O(1, 1) →

↪→

O(Σ ) πO ↓ Σ

← GL(2, R)

L(Σ )

↓ πL Σ

(86)

be the subbundle of Lorentzian frames tangent to (Σ , ι∗ Fθ ). Let O+ (Σ ) be a connected component of O(Σ ). To consider the πO+

Schmidt distance function we shall work with the principal bundle O++ (1, 1) → O+ (Σ ) → Σ with πO+ = πO |O+ (Σ ) rather than (86). Let

( ) ψ T ∈ C ∞ T ∗ (L(Σ )) ⊗ gl(2, R) ,

( ) ΘΣ ∈ C ∞ T ∗ (L(Σ )) ⊗ R2 ,

be respectively the Levi-Civita connection 1-form of the Lorentzian [by (ii) in Lemma 8] ι∗ Fθ and the canonical 1-form

ΘΣ ,u = u−1 ◦ (du πL ),

u ∈ L(Σ ).

Clearly ψ is reducible to a connection 1-form T

( ) ψ T ∈ C ∞ T ∗ (O+ (Σ )) ⊗ o(1, 1) . Let GΣ = ψ T · ψ T + ΘΣ · ΘΣ be the Schmidt metric on L(Σ ). The same symbol GΣ denotes the induced Riemannian metric on O+ (Σ ). Let dΣ : O+ (Σ ) × O+ (Σ ) → [0, +∞) be the distance function associated to GΣ and let O+ (Σ ) be the Cauchy completion of O+ (Σ ) with respect to dΣ . We set as usual

Σ = O+ (Σ )/O++ (1, 1),

˙ = Σ \ Σ. Σ

We identify O(1, 1) and O(2) with subgroups of O(1, 3) i.e. O(1, 1) ≈

O(1, 1) 0

(

0 I2

)

(

,

O(p) ≈

)

I1,1 0

0 , O(2)

where I2 is the identity matrix of order 2 and I1,1 = diag (−1, 1). Let us set O(M)|Σ = {v ∈ O(M) : ΠO (v ) ∈ Σ } where ΠO : O(M) → M is the projection. Let O(1, 1) × O(2) → O(M, Σ ) → Σ be the principal bundle of adapted frames i.e. O(M, Σ ) consists of all v ∈ O(M)|Σ such that v : R4 → Tp (M) maps R2 × {0} onto Tp (Σ ). We shall need the bundle map hT : O(M) → O(Σ ),

hT (v ) = v ⏐R2 ×{0} .

⏐

Let O(2) → O(T (Σ )⊥ ) ≈ O(M, Σ )/O(1, 1) → Σ be the principal bundle of all normal frames v : R2 → T (Σ )⊥ p with p ∈ Σ and let us consider the bundle map h⊥ : O(M, Σ ) → O(T (Σ )⊥ ),

h⊥ (v ) = v ⏐{0}×R2 .

⏐

We collect the various bundles and bundle maps introduced so far in the following commutative diagrams

O(Σ ) =

O(1, 1)

O(1, 1) × O(2)

↓ O(M, Σ )

↓

O(2)

hT

O(1, 3)

↓

↓ i

↓Π Σ

O(M)|Σ

↓ j

↓Π =

Σ

= O(1, 3)

−→

O(M)

↓Π ι

↪→

O(M, Σ ) O(1, 1)

↓ Π⊥

Σ

=

−→

−→

↓Π

O(1, 1) × O(2) O(M, Σ )

↓

h⊥

O(M, Σ )

←−

↓ ΠT Σ

O(2)

M

Σ

= O(T (Σ )⊥ )

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E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

where i and j are injections. If ΘΣ and Θ are the canonical 1-forms on O(Σ ) and O(M) respectively then (hT )∗ ΘΣ coincides with the restriction of Θ to O(M, Σ ) and in particular the restriction of the R4 -valued form Θ to O(M, Σ ) is R2 -valued [cf. Proposition 1.1 in [21], Vol. 2, p. 3]. Let g be the orthogonal complement of o(1, 1) ⊕ o(2) in o(1, 3) with respect to the Killing–Cartan form. Then the o(1, 1) ⊕ o(2)-component

( ) ω = i∗ j∗ ψ o(1,1)⊕o(2) of i∗ j∗ ψ is a connection 1-form on O(M, Σ ) [cf. Proposition 1.2 in [21], Vol. 2, p. 3]. We also recall [cf. Propositions 1.3 and 1.4 in [21], Vol. 2, p. 4] that: (i) hT : O(M, Σ ) → O(Σ ) maps the connection-distribution defined by ω on O(M, Σ ) onto the Levi-Civita connectiondistribution of (Σ , ι∗ Fθ ); (ii) the Levi-Civita connection 1-form ψ T is determined by (hT )∗ ψ T = ωo(1,1) ;

(87)

(iii) there is a unique connection 1-form ψ

⊥

on O(T (Σ ) ) such that ⊥

(h⊥ )∗ ψ ⊥ = ωo(2) ;

(88)

(iv) the map (h , h ) : O(M, Σ ) → O(Σ ) × O(T (Σ ) ) induces a principal bundle isomorphism O(M, Σ ) ≈ O(Σ ) + O(T (Σ )⊥ ) and T

⊥

⊥

ω = (hT )∗ ψ T + (h⊥ )∗ ψ ⊥ .

(89)

The connection 1-form ω may be used to produce the Schmidt like metric Gadt = ω · ω + Θ · Θ and therefore a bundle completion and bundle boundary

M

adt

= O+ (M, Σ )

adt

/O++ (1, 3),

∂adt M = M

adt

\ M,

adt

where O+ (M, Σ ) is the Cauchy completion of O+ (M, Σ ) with respect to the distance function dadt associated to the Riemannian metric Gadt (rather than i∗ dG i.e. the distance function induced on O+ (M, Σ ) by dG ). We refer to ∂adt M as the adapted bundle boundary of (M, Fθ ). H. Friedrich examined (cf. [22]) b-boundary constructions relying on arbitrary G-structures [with G ⊂ GL(4, R), a Lie subgroup]. By a result in [22] the same b-boundary is produced provided that ψ is reducible to a connection 1-form on the given G-structure. Since i∗ j∗ ψ is not a connection 1-form on O(M, Σ ) it follows ˙ and ∂adt M are in general logically unrelated objects. Our further constructions will require a pair of globally defined that M cross sections sT : Σ → O(Σ ),

s⊥ : Σ → O(T (Σ )⊥ ),

whose existence is postulated. For instance if Σ = π −1 (Γw0 ) then sT (p) : R2 → Tp (Σ ),

s⊥ (p) : R2 → T (Σ )⊥ p ,

sT (p)(e0 ) = (dγ ↑ (ϕ ) Rζ )γ˙ ↑ (ϕ ), s⊥ (p)ej+1 = Nj (p),

sT (p)e1 = Sp ,

j ∈ {1, 2},

for any p = γ (ϕ ) · ζ ∈ Σ . Our main tool in the present section is the map ↑

j : O(Σ ) → O(M, Σ ), j(u)eα = u(eα ),

j(u) : R4 → Tp (M),

j(u)ei+1 = s⊥ (p)ei+1 , α ∈ {0, 1}, i ∈ {1, 2},

for any u ∈ O(Σ )p and any p ∈ Σ . Lemma 9. Let Σ be a Lorentzian surface in (M, Fθ ) endowed with two globally defined sections sT : Σ → O(Σ ),

s⊥ : Σ → O(T (Σ )⊥ ).

If j is an isometry of (O(Σ ), GΣ ) into (O(M, Σ ), Gadt ) then: (i) j is uniformly continuous as a map of metric spaces (O+ (Σ ), dΣ ) and (O+ (M, Σ ), dadt ). adt (ii) The induced map of completions j : O+ (Σ ) → O+ (M, Σ ) is Lipschitz with Lipschitz constant L = 1 [with respect to the induced distance functions dΣ and dadt ] and equivariant i.e. it commutes with the topological group actions of O++ (1, 1) and O++ (1, 3).

E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

163

Proof. (i) For any pair of Lorentzian frames u′ , u′′ ∈ O+ (Σ ) let ΩΣ (u′ , u′′ ) [respectively ΩM (j(u′ ), j(u′′ ))] be the set of all piecewise C 1 curves in O+ (Σ ) [respectively in O+ (M, Σ )] joining u′ and u′′ [respectively j(u′ ) and j(u′′ )]. Then dadt (j(u′ ), j(u′′ )) = inf ℓGadt (C ) : C ∈ ΩM (j(u′ ), j(u′′ ))

{

}

where ℓG (C ) denotes the length of the curve C with respect to the Riemannian metric G. Let c ∈ ΩΣ (u′ , u′′ ). As j is assumed to be an isometry (i.e. j∗ Gadt = GΣ ) one has dadt (j(u′ ), j(u′′ )) ≤ ℓGadt (j ◦ c) = ℓGΣ (c) and taking the infimum over ΩΣ (u′ , u′ ) yields dadt (j(u′ ), j(u′′ )) ≤ dΣ (u′ , u′′ ). (ii) The completion O+ (Σ ) is a metric space with the distance function dΣ (u , u ) = lim dΣ (uν , u′ν ) ′

ν→∞

where {uν }ν≥1 and {u′ν }ν≥1 are Cauchy sequences in (O+ (Σ ), dΣ ) representing u and u respectively. j induces the map ′

j : O+ (Σ ) → O+ (M, Σ )

adt

,

j(u) = lim j(uν ), ν→∞

where the limit is taken with respect to the dΣ and dadt metric topologies. Let u ∈ O+ (Σ ) and B ∈ O++ (1, 1). The action of O++ (1, 1) on O+ (Σ ) is given by u · B = lim RB (uν )

(90)

ν→∞

where RB : O+ (Σ ) → O+ (Σ ) is the right translation with B. Since

j(u · B) = j(u) · B˜ ,

B˜ ≡

(

B 0

0 I2

)

,

˜ for any u ∈ O+ (Σ ) it follows that [by (90)] j(u · B) = j(u) · B.

□

If β is a connection 1-form on a given principal bundle we denote by Γ (β ) = Ker(β ) the corresponding connection distribution. Lemma 10. Let Σ ⊂ M be a Lorentzian surface endowed with the sections (sT , s⊥ ). Then for any V ∈ Γ ψ T

(

j ωo(1,1) V = 0.

(∗

)

) (91)

Consequently Γ (ψ ) = Ker(j ωo(1,1) ). T

∗

Proof. By the very definitions j is a right inverse to hT hT ◦ j = 1O(Σ ) .

(92)

Differentiating (92) one has (dj(p) hT )(dp j)Vp = Vp ∈ Γ (ψ T )p = (dj(p) hT )Γ (ω)j(p) for any p ∈ O(Σ ). Hence there is W ∈ Γ (ω)j(p) such that (dj(p) hT ) W − (dp j)Vp = 0.

[

]

(93)

Next let us apply ψ to both sides of (93) and use (87). It follows that T

( ) ωo(1,1) W − j∗ ωo(1,1) p V = 0. As W is ω-horizontal [by taking into account the direct sum decomposition o(1,)3) = o(1, 1)⊕o(2)⊕g] one has ωo(1,1) (W ) = 0 ( and we may conclude that (91) holds good. This yields Γ (ψ T )p ⊂ Ker j∗ ωo(1,1) p and the last statement in Lemma 10 follows by comparing dimensions. □ Let us observe that (again by the very definitions) h⊥ ◦ j = s ⊥ ◦ Π T .

(94)

Let a : [0, 1] → Σ be a C curve and let a : [0, 1] → O (Σ ) be a ψ -horizontal lift of a. Let A : [0, 1] → M be the curve given by A = ι ◦ a. Let AH : [0, 1] → O+ (M, Σ ) be a ω-horizontal lift of A. 1

H

+

T

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E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

Lemma 11. Let Σ be a Lorentzian surface in (M, Fθ ) with a flat normal connection (Dψ ⊥ = 0). Let sT : Σ → O(Σ ) be a given globally defined section. Let {Ni : i ∈ {1, 2}} be an orthonormal frame in T (Σ )⊥ such that ∇ ⊥ Ni = 0 for any i ∈ {1, 2} and let us consider the section s⊥ : Σ → O(T (Σ )⊥ ),

s⊥ (p)ei = Ni (p) , i ∈ {1, 2}.

Then, ( ) ∗ (i) s⊥ ψ ⊥ = 0. (ii) j : O(Σ ) → O(M, Σ ) is an isometry i.e. j∗ Gadt = GΣ . (iii) AH (t) = j(aH (t)) for any 0 ≤ t ≤ 1. adt (iv) j descends to a map of orbit spaces j : Σ → M and

˙ ⊂ ∂adt M. j Σ

( )

(95)

Proof. (i) Let p ∈ Σ and v ∈ Tp (Σ ). As v is tangent to Σ there is a C 1 curve p : (−ϵ, ϵ ) → Σ such that p(0) = p and p˙ (0) = v . Let us set b(t) = s⊥ (p(t)),

|t | < ϵ.

˙ is ψ ⊥ -horizontal. Then As Ni are parallel in the normal bundle the tangent vector b(t) Ker ψ ⊥

(

) b(t)

˙ = (dp(t) s⊥ )p˙ (t) ∋ b(t)

yields ⊥ 0 = ψb(t) (dp(t) s⊥ )p˙ (t) = (s⊥ )∗ ψ ⊥

[

] p(t)

p˙ (t)

and in particular for t = 0 (s⊥ )∗ ψ ⊥ v = 0.

[

]

(ii) Using (89), (92), (94) and (i) in Lemma 11, one has

( )∗

j ∗ ω = j ∗ hT

( )∗ ( )∗ ( )∗ ψ T + j∗ h⊥ ψ ⊥ = ψ T + Π T s⊥ ψ ⊥ = ψ T .

Also, for any u ∈ O(Σ ), (j∗ Θ )u = Θj(u) ◦ (du j) =

(( T )∗ h

ΘΣ

) j(u)

◦ (du j) = ΘΣ ,hT (j(u)) (dj(u) hT )(du j)

= ΘΣ , u . Finally,

j∗ Gadt = (j∗ ω) · (j∗ ω) + (j∗ Θ ) · (j∗ Θ ) = ψ T · ψ T + ΘΣ · ΘΣ = GΣ . (iii) Let u0 ∈ O+ (Σ ) be fixed and let aH and AH be respectively the horizontal lifts of a and A issuing at u0 and j(u0 ), i.e. aH = aH ( · ; u0 ) ,

AH = AH ( · ; j(u0 )) .

Let us also consider the curve C = j ◦ aH . Then both AH and C issue at j(u0 ) and AH is ω-horizontal. To see that C and AH coincide it suffices to check that C is ωhorizontal as well. Indeed, by (89),

ωC (t) C˙ (t) =

(( T )∗ h

ψT

) C (t)

C˙ (t) +

((

)∗

h⊥ ψ ⊥

) C (t)

C˙ (t)

= ψhTT (C (t)) (dC (t) hT )C˙ (t) + ψh⊥⊥ (C (t)) (dC (t) h⊥ )C˙ (t) and by (92), (dC (t) hT )C˙ (t) = (dC (t) hT )(daH (t) j)a˙ H (t) = a˙ H (t). Similarly, using (94), (dC (t) h⊥ )C˙ (t) = (dC (t) h⊥ )(daH (t) j)a˙ H (t) = (da(t) s⊥ )(daH (t) Π T )a˙ H (t)

= (da(t) s⊥ )a˙ (t). Therefore,

ωC (t) C˙ (t) = ψaTH (t) a˙ H (t) + ψs⊥⊥ (a(t)) (da(t) s⊥ )a˙ (t).

(96)

E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

165

Now the first term in the right hand side of (96) vanishes because a˙ H (t) is ψ T -horizontal, while the second term vanishes by statement (i) in Lemma 11. Thus ωC (t) C˙ (t) = 0. This proves (iii). adt

(iv) The map j : O + (Σ ) → O+ (M, Σ ) is equivariant [by (ii) of Lemma 9] hence induces a map j : Σ → M . To see that j ˙ i.e. an orbit p = O++ (1, 1) · u maps the b-boundary of Σ into the adapted bundle boundary of M let us consider a point p ∈ Σ for some u ∈ O+ (Σ ) such that there is a C 1 curve a : [0, 1) → Σ with the following properties + and (1) limt →1− aH (t) = u in the ∫ 1O H(Σ ) −topology, (2) the improper integral 0 ∥a (t) 1 a˙ (t)∥ dt is convergent. Let A = ι ◦ a. We claim that lim AH (t) = j(u)

(97)

t →1−

in the O+ (M, Σ ) topology. Let {tν }ν≥1 ⊂ [0, 1) be a sequence such that tν → 1 as ν → ∞. Then, for any ϵ > 0, by (iii) in Lemma 11 and (ii) in Lemma 9, dadt AH (tν ) , j(u) = dadt j(aH (tν )) , j(u) ≤ dΣ aH (tν ) , u < ϵ

(

)

(

)

(

)

for any ν ≥ νϵ and some νϵ ≥ 1. Moreover, by (92), hT (AH (t)) = (hT ◦ j)aH (t) = aH (t). Therefore, the identity i j∗ Θ

(∗

) j(u)

) ( = (hT )∗ ΘΣ j(u) ,

u ∈ O(Σ ),

for u = aH (t) yields AH (t)−1 (dAH (t) Π ) = Θ(i◦j)(AH (t)) ◦ dAH (t) (i ◦ j)

( ) = i∗ j∗ Θ AH (t) = ΘΣ , hT (AH (t)) ◦ (dAH (t) hT ) ( )( ) = ΘΣ , aH (t) ◦ (dAH (t) hT ) = aH (t)−1 daH (t) Π T dAH (t) hT , which applied to A˙ H (t) gives

˙ = aH (t)−1 daH (t) Π T a˙ H (t) = aH (t)−1 a˙ (t). AH (t)−1 A(t)

(

)

Therefore 1

∫

˙ ∥ dt = ∥AH (t)−1 A(t)

∫

1

∥aH (t)−1 a˙ (t)∥ dt < ∞

0

0

implies j(p) ∈ ∂adt M.

□

7. Principal s1 -bundles over circles Γw0 To apply Theorem 5 to the total space Σ of a principal S 1 -subbundle over a circle Γw (with |f (w )| < µ) one needs to compute its normal curvature (and show that R⊥ = 0). This amounts to the calculation of dλ where λ ∈ Ω 1 (Σ ) is the differential 1-form determined by (99). Using Weingarten’s formula [cf. (98)] the calculation of the covariant derivative ∇ ⊥ N1 may be performed by computing the covariant derivative ∇ M N1 (with respect to the Levi-Civita connection of M) followed by projection on N2 , provided that explicit smooth extensions of Ni to tangent vector fields defined on the whole of π −1 (U) may be produced. This is indeed the case, as shown in the sequel. Let U ⊂ M be the open subset built in Section 4. The parameter ϕ − f (w ) may be thought of as the function

( π) φ : U → 0, , 2

φ (x) = ϕ (r , w) − f (w),

for any x = (r ei ϕ (r ,w) , w ) ∈ U. Let

⏐ ⏐ { } ∂0 W = (z , w) ∈ C2 : ⏐z − eif (w) ⏐ = 1, |f (w)| ≤ µ be the boundary of the worm domain with the ‘‘caps’’ removed. Note that A ⊂ ∂0 W . Let us set M0 = ∂0 W \ A so that U0 = (rei ϕ (r ,w) , w ) ∈ U : |f (w )| < µ

{

}

is an open subset in M0 . Note that Γw ⊂ U0 for any |f (w )| < µ. Let us consider the function g : U → R,

g(x) =

|w| cos φ (x)

, x ∈ U,

π2 (x) = w,

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E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

where π2 : C2 → C is the projection π2 (z , w ) = w . Moreover we shall need the functions hj : M → R, j ∈ {1, 2}, given by 1

w

e−i f (w) = h1 (x) + i h2 (x) ,

for any x ∈ M with π2 (x) = w . Let g π = g ◦ π and hπj = hj ◦ π : M → R be their vertical lifts. The upper script π will be omitted when convenient. We consider the vector fields Nj ∈ X(π −1 (U)) given by N1 = g π X ↑ + hπ1 S ,

(

N2 = g π Y ↑ − hπ2 S ,

)

(

)

so that point-wise restriction to Σ = π −1 (Γw0 ) gives the orthonormal frame {N1 , N2 } in the normal bundle T (Σ )⊥ considered in Lemma 8. Let ∇ M be the Levi-Civita connection of M and let us consider the differential 1-form λ ∈ Ω 1 (π −1 (U)) defined by

( ) λ(V ) = Fθ ∇ M V N1 , N2 for any V ∈ X(π −1 (U)). The same symbol λ denotes the pullback ι∗ λ ∈ Ω 1 (Σ ). We recall the Weingarten formula (cf. e.g. [23]) ⊥ ∇M V N = −AN V + ∇ V N

(98)

for any V ∈ X(Σ ) and any N ∈ C ∞ (T (Σ )⊥ ). Here AN and ∇ ⊥ are respectively the Weingarten, or shape, operator AN (associated to the normal section N) and the normal connection. The point-wise restriction of {N1 , N2 } to Σ is Fθ -orthonormal hence (by Weingarten’s formula)

∇ ⊥ N1 = λ ⊗ N2 ,

∇ ⊥ N2 = −λ ⊗ N1 .

(99)

In particular, if ⊥ ⊥ R⊥ (V , W ) = ∇ ⊥ V , ∇ W − ∇ [V , W ]

[

]

is the curvature of the normal connection then R⊥ (V , W )N1 = 2(dλ)(V , W )N2 ,

R⊥ (V , W )N2 = −2(dλ)(V , W )N1 .

We wish to compute the forms λ and dλ. To this end we recall that (cf. e.g. Lemma 2 in [16], p. 083504-26) W ↑ = (∇V W )↑ − (dθ )(V , W )T ↑ ∇M V↑ − [A(V , W ) + (dσ )(V ↑ , W ↑ )]S , M ↑ ∇ V ↑ T = (τ V + Φ V ) ↑ , V ↑ = (∇T V + Φ V )↑ + 2(dσ )(V ↑ , T ↑ )S , ∇M T↑

(100)

↑ ↑ S = ∇M ∇M S V = (JV ) , V↑

T ↑ = X↑ , ∇M T↑

∇M S S = 0,

↑ M ∇M S T = ∇ T ↑ S = 0,

for any V , W ∈ H(M), where ∇ is the Tanaka–Webster connection of (M , θ ). Also, Φ : H(M) → H(M) is given by Gθ (Φ V , W ) = (dσ )(V ↑ , W ↑ ) and X ∈ H(M) is determined by Gθ (X, V ) = 2(dσ )(T ↑ , V ↑ ). Lemma 12. The 1-form λ ∈ Ω 1 (π −1 (U)) is given by

λ(S) =

1 2

g 2 g11 ,

λ(X ↑ ) = g 2

{

λ(Y ) = −g ↑

2

1

}

h1 g11 + Gθ (∇X X , Y ) ,

2{

1 4

h2 g11 + Gθ (X , ∇Y Y )

}

λ(T ) = g Gθ (∇T X + Φ X , Y ), 2

↑

Consequently

λ(Z ↑ ) =

1 2

g 2 g11

( h1 +

i 2

) h2

+ g 2 {Gθ (∇X X , Y ) + i Gθ (X , ∇Y Y )} .

(101)

E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

167

Moreover, (101) yield (dλ)(S , T ↑ ) = − (dλ)(S , X ↑ ) = − (dλ)(S , Y ↑ ) = −

1 4 1 4 1 4

T g 2 g11 ,

(

)

X g 2 g11 ,

(

)

(102)

Y g 2 g11 ,

)

(

Consequently, (dλ)(S , Z ↑ ) = −

1 2

Z g 2 g11 .

(

)

Proof. The calculation of (dλ)(T ↑ , X ↑ ), (dλ)(T ↑ , Y ↑ ) and (dλ)(X ↑ , Y ↑ ) may be performed along the same lines, yet will not be required in the sequel. To prove Lemma 12 note that Gθ (X , Y ) = 0,

∥X ∥2 = ∥Y ∥2 =

1 2

g11 .

Moreover, (100) and JX = Y yield ↑ ↑ ∇M S X = Y ,

hence π ↑ π 2 2 λ(S) = Fθ (∇ M S N1 , N2 ) = g Fθ (Y , N2 ) = (g ) ∥Y ∥

leading to the first identity in (101). The proof of the remaining identities in (101) is similar and therefore omitted. Moreover, note that by (100) we have S , T ↑ = S , X ↑ = S , Y ↑ = 0.

[

]

[

]

[

]

Then, by (101), 2 (dλ)(S , T ↑ ) = S(λ(T ↑ )) − T ↑ (λ(S)) = −

1 2 1

2 (dλ)(S , X ↑ ) = S(λ(X ↑ )) − X ↑ (λ(S)) = − 2 (dλ)(S , Y ↑ ) = S(λ(Y ↑ )) − Y ↑ (λ(S)) = −

2 1 2

T (g 2 g11 ) ◦ π , X (g 2 g11 ) ◦ π, Y (g 2 g11 ) ◦ π.

This proves the lemma. □ Lemma 13. The open set U0 ⊂ M0 is foliated by the circles

{Γw : |f (w)| < µ}. If Γw is the leaf passing through the point x = (r ei ϕ (r ,w) , w ) ∈ U0 then x = γw (ϕ ) for the value ϕ = ϕ (r , w ) of the parameter. Then g11 (x) =

2 g(x)2

.

(103)

Proof. One has x = γw (ϕ ) if and only if ϕ = ϕ (r , w ). Moreover [by (42)] g11 (γw (ϕ )) =

2

|w|2

cos2 [ϕ − f (w )] =

2 g(γw (ϕ ))2

.

Lemma 14. For any |f (w )| < µ and any 0 < ϕ − f (w ) <

□ π 2

(dλ)γ ↑ (ϕ ; p0 ·ζ ) Sγ ↑ (ϕ ; p0 ·ζ ) , γ˙ (ϕ ; p0 · ζ ) = 0.

(

↑

)

(104)

Consequently if Σ = π −1 (Γw ) then the normal connection of the immersion ι : Σ ↪→ M is flat (i.e. R⊥ = 0). Proof. Let p0 ∈ π −1 (U0 ) and ζ ∈ S 1 . To apply Lemma 12 one needs to extend γ˙ ↑ to a smooth vector field defined on the whole of π −1 (U0 ). A smooth extension is V = 2T − g 2 (h1 X − h2 Y ) ∈ X(π −1 (U0 )).

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E. Barletta et al. / Journal of Geometry and Physics 120 (2017) 142–168

Indeed, by Lemma 7, ↑

γ˙ ↑ (ϕ ; p0 · ζ ) = Vγ ↑ (ϕ ; p

0 ·ζ )

because both sides above are σ -horizontal and project on γ˙ (ϕ ). On the other hand, by Lemma 12 and (103) in Lemma 13, 2(dλ)(S , V ↑ ) = −T g 2 g11 +

(

)

1 2

g 2 h1 X g 2 g11 − h2 Y g 2 g11

{

(

)

(

)}

= 0,

implying that ι∗ dλ = 0. □ Proof of Theorem 6. Let {N1 , N2 } be the Fθ -orthonormal frame in T (Σ )⊥ provided by Lemma 8. To produce a new frame νi = fij Nj which is parallel in the normal bundle (∇ ⊥ νi = 0) one solves the linear PDE’s system dfi1 = fi2 λ,

dfi2 = −fi1 λ,

(105)

whose integrability condition is dλ = 0 or equivalently R = 0. By Lemma 14 the real surface Σ = π (Γw ), |f (w )| < µ, has a flat normal connection. Let then {ν1 , ν2 } be a solution to (105) and let s⊥ be the corresponding section in O(T (Σ )⊥ ). Then (s⊥ )∗ ψ ⊥ = 0 by (i) in Lemma 11. The immersion j : O(Σ ) → O(M, Σ ) is then isometric [cf. (ii) in Lemma 11]. Then −1

⊥

adt

Lemma 9 applies so that j induces a map of completions j : O+ (Σ ) → O+ (M, Σ ) . According once again to Lemma 9 this is adt ˙ into the adapted equivariant hence gives rise to a map j : Σ → M . Finally [by (iv) in Lemma 11] j maps the b-boundary Σ bundle boundary ∂adt M. □ References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

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