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Flat centroaffine surfaces with parallel second fundamental form in R4
Applied Mathematics and Computation 243 (2014) 775–788
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Flat centroaffine surfaces with parallel second fundamental form in R4 Yun Yang, Yanhua Yu ⇑, Huili Liu Department of Mathematics, Northeastern University, Shenyang, Liaoning 110004, PR China
a r t i c l e
i n f o
Keywords: Centroaffine surfaces Curvature Pick invariant The second fundamental form
a b s t r a c t By solving certain partial differential equations, we obtain some classification results for flat centroaffine surfaces in R4 , under the condition of parallel second fundamental form and vanishing Pick invariant. We also get some examples for flat centroaffine umbilical surface with parallel second fundamental form. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction The main purpose of affine differential geometry, as we know, is to study the properties of submanifolds M n of Rm that are invariant under the group of all affine transformations [9,3]. The classical theory for affine hypersurfaces was developed by Blaschke and his school [1], and has been reorganized in the last 20 years as geometry of affine immersions [8]. Similarly, the centroaffine differential geometry is the study of the properties of submanifolds that are invariant under the centroaffine transformation group, which is the subgroup of the affine transformation group that keeps the origin invariant. In centroaffine differential geometry, the theory of hypersurfaces has a long history. The notion of centroaffine minimal hypersurfaces was introduced by Wang [11] as extremals for the area integral of the centroaffine metric. The third author of this paper got the classification of surfaces in R3 which are both centroaffine-minimal and equiaffine-minimal [4], for hypersurfaces, see [7]. See also [13,14,16] for the classification results about centroaffine translation surfaces and centroaffine ruled surfaces in R3 . However, relatively little has been achieved in the study of centroaffine immersions with higher codimensions. For a centroaffine immersion into the affine space, the position vector yields its first canonical transversal vector field. A standard method of choosing a second one was proposed in 1950 by Lops˘ic (see Walter [10]). Reorganizing geometry of equi-centroaffine immersions of codimension two, Nomizu and Sasaki [8] took the prenormalized Blaschke normal field as the second canonical transversal vector field, and using this structure Furuhata [2] proved that an equi-centroaffine immersion is minimal if and only if the trace of the affine shape operator with respect to the prenormalized Blaschke normal field vanishes identically. On the other hand, the third author of this paper gave another structure to study the centroaffine immersions of codimension two [5,6]. He defined the centroaffine metric g of centroaffine immersions with codimension two in a new way and then chose Dg x as the second transversal vector field, where Dg denotes the Laplacian of g. According to this structure and using the efficacious moving frames method, the authors of this paper compared these different normalizations and defined the minimal centroaffine immersions of codimension two [12]. Recently, the authors of this paper also obtained some classification results for flat centroaffine surfaces in R4 with the condition of degenerate second fundamental form and vanishing Pick invariant [15]. In this paper, we further consider the centroaffine invariants of centroaffine
⇑ Corresponding author. E-mail addresses: [email protected] (Y. Yang), [email protected] (Y. Yu), [email protected] (H. Liu). http://dx.doi.org/10.1016/j.amc.2014.06.062 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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immersion x : M ¼ Mn ! Rnþ2 ðn P 2Þ, and find that the parallel property of the second fundamental form and umbilical property for centroaffine immersions of codimension two are invariant under centroaffine transformation. Then by solving certain systems of partial differential equations, we obtain a complete classification of flat centroaffine surfaces with parallel second fundamental form and vanishing Pick invariant in R4 . At the same time, we also get some examples for flat centroaffine umbilical surface with parallel second fundamental form. This paper is organized as follows. In Section 2 we recall the basic theory and notions for centroaffine immersions of codimension 2 with normalization fx; Dg xg. Note, there are some new definitions. In Section 3 we consider centroaffine surfaces with rh ¼ 0 in R4 , and then get a complete classification ~ h ¼ 0 in R4 , and also get a for flat surfaces with rh ¼ 0 and J ¼ 0. In Section 4 we talk about centroaffine surfaces with r ~ complete classification for flat surfaces with rh ¼ 0 and J ¼ 0. Furthermore, some flat centroaffine umbilical surfaces with parallel second fundamental form are obtained in Sections 3 and 4. Finally, in Section 5, we will give some more details for centroaffine surfaces with parallel second fundamental form appearing in Sections 3 and 4. 2. Centroaffine immersions in Rnþ2 We will recall the basic centroaffine theory in Rnþ2 . More details can be found in [12,15]. Let x : M ¼ M n ! Rnþ2 ðn P 2Þ be an oriented immersed submanifold such that xðpÞ R dxðT p MÞ for all p 2 M, and xðMÞ be not contained in a hyperplane containing the origin of Rnþ2 . Our convention for the range of indices is the following
1 6 i; j; k; . . . 6 n; n þ 1 6 a; b; c; . . . 6 n þ 2; 1 6 A; B; C; . . . 6 n þ 2 and we shall follow the usual Einstein summation convention. For any local oriented basis r ¼ fE1 ; E2 ; . . . ; En g of TM with dual basis fh1 ; h2 ; . . . ; hn g we define 2
G :¼ ½E1 ðxÞ; . . . ; En ðxÞ; x; d x ¼ Gij hi hj ;
ð2:1Þ nþ2
where ½ is the standard determinant in R that G is nondegenerate. We define
g :¼ g ij hi hj ;
, Gij :¼ ½E1 ðxÞ; . . . ; En ðxÞ; x; Ei Ej ðxÞ. G is a symmetric 2-form and we assume
1
g ij ¼ j det Gpq jnþ2 Gij :
ð2:2Þ
Then g is independent of the choice of the basis r and thus a globally defined symmetric 2-form. From Eq. (2.1) we know that the conformal class of g is a centroaffine invariant [5,6]. Definition 2.1. g is called the centroaffine metric of centroaffine immersion x : M ! Rnþ2 . Remark. x is called a nondegenerate centroaffine submanifold if g is nondegenerate and x is definite or indefinite if g is definite or indefinite, respectively. n o D x Definition 2.2. Let Dg denote the Laplacian of g. x; ng is called the centroaffine normalization of centroaffine immersion x : M ! Rnþ2 . Let U be a connected open subset of manifold M. If no confusion is possible, we will identify U with its image xðUÞ and T u M n o D x with x ðT u MÞ. We denote the canonical flat connection on Rnþ2 by D. With respect to the frame E1 ðxÞ; . . . ; En ðxÞ; x; ng , the structure equations for arbitrary tangential vector fields X; Y 2 XðUÞ can be written in the form [12]:
Dg x DY X ¼ rY X þ hðX; YÞx þ gðX; YÞ ; n Dg x ¼ SðXÞ þ qðXÞx: DX n
ð2:3Þ ð2:4Þ
A standard proof shows that r is a torsion free affine connection, h are symmetric bilinear forms, shape operator S is a (1, 1)-tensor field and q are 1-forms defined on U[12]. If we denote by Rr the curvature tensor of r, i.e., for arbitrary tangent vector fields X; Y; Z 2 XðUÞ; Rr ðX; YÞZ ¼ rX rY Z rY rX Z r½X;Y Z, then the centroaffine integrability conditions are
Rr ðX; YÞZ ¼ gðY; ZÞSðXÞ hðY; ZÞX gðX; ZÞSðYÞ þ hðX; ZÞY;
ð2:5Þ
ðrX gÞðY; ZÞ ¼ ðrY gÞðX; ZÞ;
ð2:6Þ
ðrX SÞðYÞ þ qðXÞY ¼ ðrY SÞðXÞ þ qðYÞX;
ð2:7Þ
hðX; SðYÞÞ hðY; SXÞ ¼ ðrX qÞðYÞ ðrY qÞðXÞ;
ð2:8Þ
gðX; SðYÞÞ gðY; SðXÞÞ ¼ 0:
ð2:9Þ
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~ ¼ fC ~ k g be the induced connection and the Levi–Civita connection of centroaffine In local coordinates, let r ¼ fCkij g and r ij metric g, and then we define the Fubini-Pick form C by
~k; C kij :¼ Ckij C ij C ijk :¼
ð2:10Þ
g il C ljk :
ð2:11Þ
The Pick invariant is defined by
J¼
1 g ij C lik C klj : nðn 1Þ
In [12] we obtained that C kij is symmetric for i and j and C ijk is symmetric for i; j and k. Furthermore we have
g ij C kij ¼ 0;
g ij hij ¼ 0;
C iij ¼ 0:
ð2:12Þ
~ij ¼ g Sl and H ~ij Þ, ~ ¼ ðh Introducing now h il j
H¼
~ trg H trS ¼ n n
ð2:13Þ
is called the centroaffine mean curvature of centroaffine immersion x : M n ! Rnþ2 , and the centroaffine immersion x is centroaffine minimal if and only if H ¼ 0 [12]. Certainly, we also have the centroaffine theorema egregium [15]
v¼
~ jk g jk R R ¼ ¼ J H; nðn 1Þ nðn 1Þ
ð2:14Þ
where R is the scalar curvature, v is the normalized scalar curvature of the metric g. Here we can give the following two definitions for centroaffine immersion of codimension two. Definition 2.3
L ¼ hij dxi dxj is call the second fundamental form of centroaffine immersion x : M ! Rnþ2 . Then centroaffine immersion x is said to have ~ h ¼ 0. parallel second fundamental form if rh ¼ 0 or r Definition 2.4. A surface in R4 is called centroaffine umbilical if the shape operator S has only one eigenvalue. : M ! Rnþ2 are centroaffine equivalent, then g ¼ kg, where k is constant, and it Remark. If two centroaffine immersion x; x ~ is easy to verify rh ¼ 0; rh ¼ 0 and umbilical property are invariant under centroaffine transformation from the structure Eqs. (2.3) and (2.4).
3. Flat centroaffine surfaces with $h ¼ 0 In this section we will deduce the following theorem which is a complete classification for flat centroaffine surfaces with J ¼ 0 and rh ¼ 0 in R4 . Theorem 3.1. Let x : M ! R4 be a flat centroaffine surface with J ¼ 0 and rh ¼ 0, then x is centroaffinely equivalent to one of the following surfaces in R4 : 1. 2. 3. 4. 5. 6.
Tran
x ¼ ðu2 þ v 2 ; u; v ; 1Þ ; Tran ðcosh u; sinh u; cos v ; sin v Þ ; Tran ðcosh u; sinh u; cosh v ; sinh v Þ ; Tran ðcos u; sin u; cos v ; sin v Þ ; Tran ðcosh u cos v ; cosh u sin v ; sinh u cos v ; sinh u sin v Þ ; x ¼ v V 1 ðuÞ þ V 2 ðuÞ where V 1 ðuÞ; V 2 ðuÞ are linearly independent 1-variable vector kðuÞ lðuÞ ; kðuÞ and lðuÞ are arbitrary 1-variable functions. ðV 001 ðuÞ; V 002 ðuÞÞ ¼ ðV 1 ðuÞ; V 2 ðuÞÞ 0 c
fields
satisfying
that
Remark. From the centroaffine theorema egregium (2.14), we know flat centroaffine surfaces with vanishing Pick invariant are centroaffine minimal, so the surfaces included in Theorem 3.1 all are centroaffine minimal.
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Especially, in this section, we also find the following examples of flat centroaffine umbilical surfaces with rh ¼ 0: Tran
ðu2 þ v 2 ; u; v ; 1Þ ; Tran 1 u; v ; ;1 ; uv Tran 1 1 u2 ; sin v ; cos v ; 1 ; u u
ð3:1Þ ð3:2Þ ð3:3Þ
v V 1 ðuÞ þ V 2 ðuÞ;
ð3:4Þ V 001 ðuÞ
V 002 ðuÞ
where V 1 ðuÞ; V 2 ðuÞ are linearly independent 1-variable vector fields satisfying that ¼ kðuÞV 1 ðuÞ; ¼ lðuÞV 1 ðuÞ, and kðuÞ; lðuÞ are arbitrary 1-variable functions. Exactly, in following Section 3.1 we will get the classifications of definite centroaffine surfaces with J ¼ 0 and rh ¼ 0. Then under the same condition, the classifications of indefinite centroaffine surfaces will be obtained in Section 3.2. At the same time, the umbilical surfaces will be considered in Sections 3.1 and 3.2. 3.1. Definite centroaffine surfaces in R4 Let x : M ! R4 be a centroaffine surface with definite centroaffine metric g. Hence we can choose a local basis r ¼ fE1 ; E2 g of TM such that
g ¼ ex ðdu2 þ dv 2 Þ;
ð3:5Þ
where ¼ 1. Obviously, g 11 ¼ g 22 ¼ From Eq. (2.12) we get
ex ;
4
g 12 ¼ g 21 ¼ 0, and fE1 ðxÞ; E2 ðxÞ; x; Dg xg forms a local moving frame for R along M.
g ij hij ¼ ex ðh11 þ h22 Þ ¼ 0;
ð3:6Þ
that is
h11 þ h22 ¼ 0:
ð3:7Þ
By a direct calculation, we know that the coefficients of the Levi–Civita connection of the centroaffine metric g are
~1 ¼ C 11 ~1 ¼ C 12
xu 2
;
~ 2 ¼ xv ; C 11
;
~2 ¼ C 12
xv 2
xu
~1 ¼ C 22
2
xu
ð3:9Þ
;
2
~2 ¼ C 22
;
ð3:8Þ
2
xv 2
ð3:10Þ
:
Again from Eq. (2.12) it is easy to find
C111 þ C212 ¼ xu ;
C112 þ C222 ¼ xv ;
C111 þ C122 ¼ 0;
C211 þ C222 ¼ 0:
ð3:11Þ
Then the Pick invariant
J¼
ex 2
2
2
ððxu 2C111 Þ þ ðxv 2C222 Þ Þ:
ð3:12Þ
By the flatness assumption of x we can choose the coordinates such that x 0, which leads to
C111 ¼ C212 ¼ C122 ¼ aðu; v Þ; 2 22
C
¼ C 2
1 12
¼ C
2 11
¼ bðu; v Þ;
2
J ¼ 2ða þ b Þ;
ð3:13Þ ð3:14Þ ð3:15Þ
where the notations a; b are mainly for simplification. According to the assumption rh ¼ 0 and Eq. (3.7), a direct calculation yields
@h11 ¼ 2ðah11 bh12 Þ; @u @h11 ¼ 2ðah11 bh12 Þ; @u @h11 ¼ 2ðbh11 ah12 Þ; @v @h11 ¼ 2ðbh11 ah12 Þ; @v
ð3:16Þ ð3:17Þ ð3:18Þ ð3:19Þ
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Y. Yang et al. / Applied Mathematics and Computation 243 (2014) 775–788
@h12 ¼ 0; @u @h12 ¼ 0: @v
ð3:20Þ ð3:21Þ
It immediately follows that h12 is constant and
a b b
a
h11 h12
¼ 0:
ð3:22Þ 2
Surely, the direct result of Eq. (3.22) is a2 þ b ¼ 0 or h11 ¼ h12 ¼ 0. In the following we consider them respectively. 2
Case 1. a2 þ b ¼ 0. 2 According to Eq. (3.15), it is clear that a2 þ b ¼ 0 is equivalent to J ¼ 0, and also a ¼ b ¼ 0. Again from Eqs. (3.16)–(3.21), we can find hij are constants, where i; j 2 f1; 2g. Therefore, the structure equations can be changed as
xuu ¼ h11 x þ
Dg x ; 2
ð3:23Þ
xuv ¼ h12 x;
ð3:24Þ
Dg x ; xvv ¼ h22 x þ 2 D x g ¼ h22 xu þ h12 xv ; 2 u D x g ¼ h12 xu h11 xv : 2 v
ð3:25Þ ð3:26Þ ð3:27Þ
If h12 ¼ 0, we can get a system of partial differential equations from Eqs. (3.23) and (3.24),
xuv ¼ 0;
ð3:28Þ
xuu xvv ¼ ðh11 h22 Þx:
ð3:29Þ
The condition h11 h22 ¼ 0 implies the surface is centroaffinely equivalent to [15] Tran
ðu2 þ v 2 ; u; v ; 1Þ
ð3:30Þ
:
On the other hand, the condition h11 h22 – 0 deduces the surface is centroaffinely equivalent to [6] Tran
ðcosh u; sinh u; cos v ; sin v Þ
:
ð3:31Þ
Obviously, these two surfaces are (1) and (2) in Theorem 3.1. Tran
Remark. Eq. (3.7) and h11 h22 ¼ 0 can lead to h11 ¼ h22 ¼ 0, so it is easy to see the surface ðu2 þ v 2 ; u; v ; 1Þ is a centroaffine umbilical surface with shape operator S ¼ 0. If h12 – 0 and h11 h22 ¼ 0, which implies h11 ¼ h22 ¼ 0 from Eq. (3.7), then parameter transformation ¼ u þ v; v ¼ u v changes the structure Eqs. (3.23)–(3.25) to u
xuu xv v ¼ h12 x;
ð3:32Þ
xuv ¼ 0:
ð3:33Þ
This surface, indeed, is centroaffinely equivalent to (3.31). If h12 – 0 and h11 h22 – 0, which means h11 ¼ h22 – 0 from Eq. (3.7), we can take
0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 h h11 A 11 ¼@ 1þ u þ v; u h12 h12 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 h11 h11 A @ v ¼ þ 1þ u þ v: h12 h12
ð3:34Þ
ð3:35Þ
Then Eqs. (3.23)–(3.25), that is, the system
xuv ¼ h12 x;
ð3:36Þ
xuu xvv ¼ 2h11 x;
ð3:37Þ
xuu xvv 2
h11 xu v ¼ 0 h12
ð3:38Þ
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Y. Yang et al. / Applied Mathematics and Computation 243 (2014) 775–788
can be changed to
xuv ¼ 0; 0 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 2 h h h h11 A 11 11 11 @ Axuu þ @ xv v ¼ h12 x: 1þ þ 1þ h12 h12 h12 h12
ð3:39Þ ð3:40Þ
By solving this system of partial differential equations, we know this surface is also centroaffinely equivalent to (3.31). 2
Case 2. a2 þ b – 0. 2 From Eq. (3.22), we know a2 þ b – 0 implies h11 ¼ h12 ¼ 0. Together with Eq. (3.7), it is direct to see hij ¼ 0, where i; j 2 f1; 2g. Then the structure equations can be rewritten as
xuu ¼ axu bxv þ
Dg x ; 2
ð3:41Þ
xuv ¼ bxu axv ;
ð3:42Þ
Dg x xvv ¼ axu þ bxv þ ; 2 D x 2 g ¼ ð2a2 þ 2b þ au bv Þxu þ ðav bu Þxv ; 2 u D x 2 g ¼ ðav bu Þxu þ ð2a2 þ 2b þ bv au Þxv : 2 v
ð3:43Þ ð3:44Þ ð3:45Þ
Here we assume x is a centroaffine umbilical surface, which implies the shape operator S has only one eigenvalue kðu; v Þ. From Eqs. (3.44) and (3.45), we have
ðSij Þ
¼
ð2a2 þ 2b þ au bv Þ
2
av þ bu
av þ bu
ð2a2 þ 2b þ bv au Þ
2
! :
ð3:46Þ
It immediately follows that
8 > < au ¼ bv ; av ¼ bu ; > : 2 2 2a þ 2b ¼ k: Then by
with u and
Dg x 2 uv
au av
¼
Dg x 2
vu
ð3:47Þ we get ku ¼ kv ¼ 0, i.e., k is nonzero constant. Differentiating both sides of the above last equation
v respectively, we have bu bv
a b
¼ 0:
ð3:48Þ
2
Together with a2 þ b – 0, the first two equations of (3.47) generate
a2u þ a2v ¼ 0: It immediately deduces a and b are constants. Then from Eqs. (3.41) and (3.42), we obtain
ðeauþbv xÞuv ¼ abðeauþbv xÞ;
ð3:49Þ 2
ðeðauþbv Þ xÞuu ðeðauþbv Þ xÞvv ¼ ða2 b Þðeðauþbv Þ xÞ:
ð3:50Þ
Remark. Obviously, if x is a solution of Eqs. (3.49) and (3.50), so is x þ V, where V is an arbitrary constant vector. In the following, we will give a solution for Eqs. (3.49) and (3.50) get a flat centroaffine umbilical surface with parallel second fundamental form. (1) If ab ¼ 0, then x ¼ eðauþbv Þ ðV 1 ðuÞ þ V 2 ðv ÞÞ, where V 1 ðuÞ; V 2 ðv Þ are 1-variable vector fields with respect to u and v respectively. Without loss of generality, we may assume that a ¼ 0 and x ¼ ebv ðV 1 ðuÞ þ V 2 ðv ÞÞ. Combining of Eqs. (3.41) and (3.43) leads to xuu xvv ¼ 2bxv , so it follows that
V 1 ðuÞ ¼ e
pffiffi 3bu bv
3 þ e V 2 ðv Þ ¼ e V
3bv
pffiffi 3bu
V 2 V 0 ; V 4 þ V 0 ;
V 1 þ e
ð3:51Þ ð3:52Þ
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Y. Yang et al. / Applied Mathematics and Computation 243 (2014) 775–788
i ; i 2 f0; 1; 2; 3; 4g are constant vectors and V 1 ; V 2 ; V 3 ; V 4 are linearly independent. Thus the surface is centroaffwhere V inely equivalent to
ðeuþv ; euv ; e2u ; 1Þ
Tran
ð3:53Þ
:
By a parameter transformation, this surface is equivalent to
Tran 1 u; v ; ;1 ; uv
ð3:54Þ
which is an example of flat centroaffine umbilical surfaces with parallel second fundamental form. 2 ¼ au þ bv ; v ¼ au bv , we can obtain the same surface as Eq. (2) If ab – 0 and a2 ¼ b , by using variable substitution u (3.53). 3.2. Indefinite centroaffine surfaces in R4 Let x : M ! R4 be a centroaffine surface with indefinite centroaffine metric g. By choosing the asymptotic local basis
r ¼ fE1 ; E2 g of TM such that g ¼ 2ex ðdudv Þ ¼ ex du dv þ ex dv du; we get g 11 ¼ g 22 ¼ 0; g 12 ¼ g 21 ¼ Eq. (2.12) we get
ex ,
ð3:55Þ 4
and fE1 ðxÞ; E2 ðxÞ; x; Dg xg forms a local moving frame for R along M. Again from
g ij hij ¼ ew ðh12 þ h21 Þ ¼ 0;
ð3:56Þ
that is,
h12 þ h21 ¼ 2h12 ¼ 0:
ð3:57Þ
The structure equations can be rewritten as
xuu ¼ C111 xu þ C211 xv þ h11 x; Dg x xuv ¼ C112 xu þ C212 xv þ ex ; 2
ð3:58Þ
xvv ¼ C122 xu þ C222 xv þ h22 x; Dg x ¼ S11 xu S21 xv þ q1 x; 2 u Dg x ¼ S12 xu S22 xv þ q2 x: 2 v
ð3:60Þ
ð3:59Þ
ð3:61Þ ð3:62Þ
By a direct calculation, we can get the coefficients of the Levi–Civita connection of the centroaffine metric g
~ 1 ¼ xu ; C 11 ~ 1 ¼ 0; C 12 ~ 1 ¼ 0; C 22
~ 2 ¼ 0; C 11 2 ~ C12 ¼ 0; ~ 2 ¼ xv : C
ð3:63Þ ð3:64Þ ð3:65Þ
22
According to Eq. (2.12) we have
C111 ¼ xu ;
C112 ¼ C212 ¼ 0;
C222 ¼ xv :
ð3:66Þ 4
Similarly, we get the Pick invariant of indefinite centroaffine surface x in R
J ¼ ex C211 C122 :
ð3:67Þ
Then by the assumption of flatness for x, we can choose the coordinates such that x 0. Thus combing of Eqs. (3.66) and (3.67) gives
C111 ¼ C112 ¼ C212 ¼ C222 ¼ 0:
ð3:68Þ
The assumption rh ¼ 0, together with Eq. (3.57), yields
@h11 ¼ 0; @u @h22 ¼ 0; @v
@h11 ¼ 0; @v
C122 h11 ¼ 0;
@h22 ¼ 0; @u
C211 h22 ¼ 0:
ð3:69Þ ð3:70Þ
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It immediately follows that h11 and h22 are constants. Depending on J ¼ 0 and J – 0, in the following we will divide into two cases. Case 1. C211 C122 ¼ 0. From Eq. (3.67), it is clear that C211 C122 ¼ 0 is equivalent to J ¼ 0. According to the different situations of hij ; i; j 2 f1; 2g, we will have the following classifications. (1)
If h11 h22 – 0, Eq. (3.70) leads to C211 ¼ C122 ¼ 0. According to Eqs. (3.58)–(3.62), we have
xuu ¼ h11 x;
xu v ¼
Dg x ¼ h11 xv ; 2 u
Dg x ; xvv ¼ h22 x; 2 Dg x ¼ h22 xu : 2 v
ð3:71Þ
ð3:72Þ
Firstly, if h11 > 0 and h22 > 0, by solving the system of partial equation (3.71) we have
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi x ¼ e h11 uþ h22 v V 1 þ e h11 u h22 v V 2 þ e h11 u h22 v V 3 þ e h11 uþ h22 v V 4 ; where V i ; i 2 f1; 2; 3; 4g are four linearly independent constant vectors, and this surface is centroaffinely equivalent to
ðcosh u; sinh u; cosh v ; sinh v Þ
Tran
ð3:73Þ
:
Then if h11 < 0 and h22 < 0, by solving the system of partial equation (3.71) we have
x ¼ cos u cos v V 1 þ cos u sin v V 2 þ sin u cos v V 3 þ sin u sin v V 4 ; where V i ; i 2 f1; 2; 3; 4g are four linearly independent constant vectors, and this surface is centroaffinely equivalent to
ðcos u; sin u; cos v ; sin v Þ
Tran
ð3:74Þ
:
Otherwise, if h11 h22 < 0, by solving the system of partial equation (3.71), the surface is centroaffinely equivalent to
ðcosh u cos v ; cosh u sin v ; sinh u cos v ; sinh u sin v Þ
Tran
:
ð3:75Þ
These three surfaces are surfaces (3), (4) and (5) in Theorem 3.1 respectively. (2) If h11 ¼ h22 ¼ 0 and C211 ¼ C122 ¼ 0, the structure Eqs. (3.58)–(3.61), can be rewritten as
Dg x xuu ¼ xvv ¼ 0; xu v ¼ ; 2 Dg x Dg x ¼ ¼ 0: 2 u 2 v
ð3:76Þ ð3:77Þ
By solving this system of partial differential equations, this surface is centroaffinely equivalent to
ðu; v ; uv ; 1ÞTran ;
ð3:78Þ
which is included in the surface (6) of Theorem 3.1. Obviously, its shape operator S ¼ 0, so this surface is one example of flat centroaffine umbilical surfaces with rh ¼ 0. (3) If h11 ¼ h22 ¼ 0 and one of C211 and C122 is not zero, without loss of generality we can assume C211 – 0. Eqs. (3.58) and (3.62) can be rewritten as
xuu ¼ C211 xv ;
xu v ¼
Dg x @ C211 ¼ xv ; 2 u @v
Dg x ; 2
xvv ¼ 0;
ð3:79Þ
Dg x ¼ 0: 2 v
ð3:80Þ @ 2 C2
Then we can get the integrability conditions @ v 211 ¼ 0, which implies C211 ¼ kðuÞv þ lðuÞ, where kðuÞ; lðuÞ are arbitrary 1-variable functions and k2 ðuÞ þ l2 ðuÞ – 0. Therefore, the surface is centroaffinely equivalent to
v V 1 ðuÞ þ V 2 ðuÞ; where V 1 ðuÞ; V 2 ðuÞ are linearly independent 1-variable vector fields satisfying that V 001 ðuÞ ¼ kðuÞV 1 ðuÞ; V 002 ðuÞ ¼ lðuÞV 1 ðuÞ. Surely, this surfaces is also included in the surface (6) of Theorem 3.1. From Eq. (3.80) we know that this surface is also one of flat centroaffine umbilical surfaces. (4) If one of h11 and h22 is not zero, without loss of generality we can assume h22 ¼ 0 and h11 – 0. From Eq. (3.70) we obtain C122 ¼ 0. Eqs. (3.58) and (3.60) can be rewritten as
xuu ¼ C211 xv þ h11 x;
xvv ¼ 0:
ð3:81Þ
Y. Yang et al. / Applied Mathematics and Computation 243 (2014) 775–788
783
@ 2 C2
Using xuuvv ¼ xvv uu , we also get the integrability conditions @ v 211 ¼ 0, which implies C211 ¼ c1 ðuÞv þ c2 ðuÞ, where c1 ðuÞ; c2 ðuÞ are arbitrary 1-variable functions. Therefore, the surface is centroaffinely equivalent to
v V 1 ðuÞ þ V 2 ðuÞ; where V 1 ðuÞ; V 2 ðuÞ are linearly independent 1-variable vector fields satisfying that
ðV 001 ðuÞ; V 002 ðuÞÞ ¼ ðV 1 ðuÞ; V 2 ðuÞÞ
c1 ðuÞ þ h11
c2 ðuÞ
0
h11
:
Exactly, this surface is also included in (6) in Theorem 3.1. Case 2. C211 C122 – 0. From Eq. (3.70), it is clear that C211 C122 – 0 implies h11 ¼ h22 ¼ 0. Then the structure equations can be rewritten as
xuu ¼ C211 xv ; Dg x xuv ¼ ; 2 1 xvv ¼ C22 xu ; Dg x @ C2 ¼ C211 C122 xu þ 11 xv ; 2 u @v Dg x @ C122 2 ¼ xu þ C11 C122 xv : 2 v @u
ð3:82Þ ð3:83Þ ð3:84Þ ð3:85Þ ð3:86Þ
Here we also assume x is a centroaffine umbilical surface, that is, the shape operator S has only one eigenvalue kðu; v Þ. By a simple calculation, we obtain
C211 C122 ¼ k;
ðC211 Þv ðC122 Þu ¼ 0:
ð3:87Þ
The integrability condition generates
@ 2 C211 @ C122 2 @k ¼ C þ ; @v 2 @u 11 @u @ 2 C122 @ C211 1 @k ¼ C þ : @u2 @ v 22 @ v
ð3:88Þ ð3:89Þ
Especially, if k is nonzero constant, from Eqs. (3.87) and (3.89), we can deduce C211 and C122 are constants. Taking the notation a :¼ C211 ; b :¼ C122 for simplification, we obtain
xuu ¼ axv ;
xvv ¼ bxu :
ð3:90Þ
Due to a; b are constants, we have
xuuv ¼ axvv ¼ abxu ;
xvv u ¼ bxuu ¼ abxv ;
ð3:91Þ
which implies
ðxuv abxÞu ¼ 0;
ðxv u abxÞv ¼ 0:
ð3:92Þ
Therefore, we get
xuv abx ¼ abV 1 ;
ð3:93Þ
where V 1 is arbitrary constant vector. Together with Eq. (3.90), it follows that
ðx V 1 Þuu ¼ aðx V 1 Þv ;
ðx V 1 Þuv ¼ abðx V 1 Þ;
ðx V 1 Þvv ¼ bðx V 1 Þu :
ð3:94Þ
Assume a ¼ b. Obviously,
x ¼ V1 þ e
aðuþv Þ
V2 þ e
2aðuþv Þ
! ! pffiffiffi pffiffiffi 3a 3a 2aðuþv Þ cos ðu v Þ V 3 þ e sin ðu v Þ V 4 2 2
ð3:95Þ
gives a solution of Eq. (3.94), where V 1 ; V 2 ; V 3 ; V 4 are four linearly independent constant vectors. By parameter transformation and centroaffine transformation, this surface is equivalent to
Tran 1 1 u2 ; sin v ; cos v ; 1 ; u u which also is an example of flat centroaffine umbilical surfaces with parallel second fundamental form.
ð3:96Þ
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~ ¼0 4. Flat centroaffine surfaces with $h ~ h ¼ 0 implies all hij are constants, where i; j 2 f1; 2g. From the structure equaSince the centroaffine surface x is flat, r ~ h ¼ 0. tions in Section 3 it is not difficult to verify the surfaces (3.1)–(3.4) are also flat centroaffine umbilical surfaces with r For definite centroaffine surfaces with J ¼ 0, we can obtain its structure equations
xuu ¼ h11 x þ
Dg x ; 2
ð4:1Þ
xuv ¼ h12 x;
ð4:2Þ
Dg x xvv ¼ h11 x þ ; 2 D x g ¼ h12 xu h11 xv ; 2 u D x g ¼ h22 xu þ h12 xv : 2 v
ð4:3Þ ð4:4Þ ð4:5Þ
Then we get equations
xuv ¼ h12 x;
ð4:6Þ
xuu xvv ¼ 2h11 x:
ð4:7Þ
According to the calculations in Case 1 of Section 3.1, we know the surface is centroaffinely equivalent to Tran
ðcosh u; sinh u; cos v ; sin v Þ
ð4:8Þ
or
ðu2 þ v 2 ; u; v ; 1Þ
Tran
ð4:9Þ
:
For indefinite centroaffine surfaces with J ¼ 0, we get its structure equations
xuu ¼ C211 xv þ h11 x; xuv ¼
ð4:10Þ
Dg x ; 2
ð4:11Þ
xvv ¼ C122 xu þ h22 x; ! Dg x @ C211 ¼ þ h11 xv þ h22 C211 x; 2 u @v ! Dg x @ C122 ¼ þ h22 xu þ h11 C122 x; 2 u @u
ð4:12Þ ð4:13Þ
ð4:14Þ
where C122 C211 ¼ 0. Without loss of generality, we may assume that C122 ¼ 0. The integrability condition gives @ 2 C211 @v 2
@ C2
¼ 0; @v11 h22 ¼ 0. If h22 ¼ 0, the surface is centroaffinely equivalent to
v V 1 ðuÞ þ V 2 ðuÞ; where V 1 ðuÞ; V 2 ðuÞ are linearly independent 1-variable vector fields satisfying that
kðuÞ ðV 001 ðuÞ; V 002 ðuÞÞ ¼ ðV 1 ðuÞ; V 2 ðuÞÞ 0
lðuÞ c
; @ C2
7kðuÞ; lðuÞ are arbitrary 1-variable functions, and c is constant. If h22 – 0, then @v11 ¼ 0 and we may assume C211 ¼ f ðuÞ, where f ðuÞ is an arbitrary 1-variable function. Then if h22 > 0, the surface is centroaffinely equivalent to
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi sinhð h22 v ÞV 1 ðuÞ þ coshð h22 v ÞV 2 ðuÞ; where V 1 ðuÞ; V 2 ðuÞ are linearly independent 1-variable vector fields satisfying that V 001 ðuÞ ¼ ðh11 þ pffiffiffiffiffiffiffi V 002 ðuÞ ¼ ðh11 h22 f ðuÞÞV 2 ðuÞ. Otherwise, if h22 < 0, the surface is centroaffinely equivalent to
sin
pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi h22 v V 1 ðuÞ þ cos h22 v V 2 ðuÞ;
pffiffiffiffiffiffiffi h22 f ðuÞÞV 1 ðuÞ and
Y. Yang et al. / Applied Mathematics and Computation 243 (2014) 775–788
785
where V 1 ðuÞ; V 2 ðuÞ are linearly independent 1-variable vector fields satisfying that
ðV 001 ðuÞ; V 002 ðuÞÞ ¼ ðV 1 ðuÞ; V 2 ðuÞÞ
! pffiffiffiffiffiffiffiffiffiffiffi h22 f ðuÞ
h11 pffiffiffiffiffiffiffiffiffiffiffi h22 f ðuÞ
h11
:
Therefore, we obtain the following theorem. ~ h ¼ 0, then x is centroaffinely equivalent to one of the Theorem 4.1. Let x : M ! R4 be a flat centroaffine surface with J ¼ 0 and r following surfaces in R4 : Tran
1. x ¼ ðu2 þ v 2 ; u; v ; 1Þ ; Tran 2. ðcosh u; sinh u; cos v ; sin v Þ ; 3. x ¼ v V 1 ðuÞ þ V 2 ðuÞ, where V 1 ðuÞ; V 2 ðuÞ are linearly independent 1-variable vector fields satisfying kðuÞ lðuÞ ; kðuÞ and lðuÞ are arbitrary 1-variable function, and c is constant; ðV 001 ðuÞ; V 002 ðuÞÞ ¼ ðV 1 ðuÞ; V 2 ðuÞÞ 0 c
that
4. x ¼ sinh v V 1 ðuÞ þ cosh v V 2 ðuÞ, where V 1 ðuÞ; V 2 ðuÞ are linearly independent 1-variable vector fields satisfying that V 001 ðuÞ ¼ ðc þ kðuÞÞV 1 ðuÞ and V 002 ðuÞ ¼ ðc kðuÞÞV 2 ðuÞ; kðuÞ is an arbitrary 1-variable functions, and c is constant; 5. sin v V 1 ðuÞ þ cos v V 2 ðuÞ, ðV 001 ðuÞ; V 002 ðuÞÞ
where V 1 ðuÞ; V 2 ðuÞ are linearly independent 1-variable vector fields satisfying c kðuÞ ; kðuÞ is an arbitrary 1-variable function, and c is nonzero constant. ¼ ðV 1 ðuÞ; V 2 ðuÞÞ kðuÞ c
that
Remark. From the centroaffine theorema egregium (2.14), we know flat centroaffine surfaces with vanishing Pick invariant are centroaffine minimal, so the surfaces included in Theorem 4.1 are centroaffine minimal. On the other hand, it is easy to find that all surfaces of Theorem 3.1 are some examples of Theorem 4.1. 5. The examples in Sections 3 and 4 In this section we will calculate the basic invariants for the surfaces appearing in Sections 3 and 4, and give their structure equations. Tran
1. For the surface x ¼ ðu2 þ v 2 ; u; v ; 1Þ
g 11 ¼ g 22
pffiffiffi ¼ 2;
, it follows from Eq. (2.2) that
g 12 ¼ g 21 ¼ 0:
ð5:1Þ
By a straightforward computation we get
pffiffiffi Tran Dg x ¼ ð 2; 0; 0; 0Þ : 2
ð5:2Þ
Thus, the structure equations can be written as
pffiffiffiDg x ; xuu ¼ xvv ¼ 2 2
ð5:3Þ
xuv ¼ 0; Dg x Dg x ¼ ¼ 0: 2 u 2 v 2. For the surface x ¼ ðcosh u; sinh u; cos v ; sin v Þ
g 11 ¼ g 22 ¼ 1;
ð5:4Þ ð5:5Þ Tran
, from Eq. (2.2) we can get the centroaffine metric
g 12 ¼ g 21 ¼ 0:
ð5:6Þ
Then we can obtain the second transversal vector field
Dg x 1 Tran ¼ ðcosh u; sinh u; cos v ; sin v Þ : 2 2
ð5:7Þ
Therefore, the structure equations are given by
1 Dg x 1 Dg x xuu ¼ x þ ; xuv ¼ 0; xvv ¼ x þ ; 2 2 2 2 Dg x 1 Dg x 1 ¼ xu ; ¼ xv : 2 u 2 2 v 2
ð5:8Þ ð5:9Þ Tran
3. For the surface x ¼ ðcosh u; sinh u; cosh v ; sinh v Þ
g 11 ¼ 1;
g 22 ¼ 1;
g 12 ¼ g 21 ¼ 0:
, from Eq. (2.2) we also get its centroaffine metric
ð5:10Þ
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Y. Yang et al. / Applied Mathematics and Computation 243 (2014) 775–788
A direct computation gives
Dg x 1 Tran ¼ ðcosh u; sinh u; cosh v ; sinh v Þ : 2 2
ð5:11Þ
Then, we have
1 Dg x 1 Dg x xuu ¼ x þ ; xuv ¼ 0; xvv ¼ x ; 2 2 2 2 Dg x 1 Dg x 1 ¼ xu ; ¼ xv : 2 u 2 2 v 2 4. For the surface x ¼ ðcos u; sin u; cos v ; sin v Þ
g 11 ¼ 1;
g 22 ¼ 1;
ð5:12Þ ð5:13Þ
Tran
, again from Eq. (2.2) it follows that
g 12 ¼ g 21 ¼ 0:
ð5:14Þ
By a straightforward computation we have
Dg x 1 Tran ¼ ðcos u; sin u; cos v ; sin v Þ : 2 2
ð5:15Þ
Then, the structure equations can be written as
1 Dg x 1 Dg x xuu ¼ x ; xuv ¼ 0; xvv ¼ x þ ; 2 2 2 2 Dg x 1 Dg x 1 ¼ xu ; ¼ xv : 2 u 2 2 v 2 5. For the surface x ¼ ðcosh u cos v ; cosh u sin v ; sinh u cos v ; cosh u sin v Þ
g 11 ¼ g 22 ¼ 0;
g 12 ¼ g 21 ¼ 1:
ð5:16Þ ð5:17Þ Tran
, it is easy to verify from Eq. (2.2) that
ð5:18Þ
A straightforward computation yields
Dg x Tran ¼ ðsinh u sin v ; sinh u cos v ; cosh u sin v ; cosh u cos v Þ : 2
ð5:19Þ
Thus, the structure equations can be represented as
Dg x ; xvv ¼ x; xuu ¼ x; xuv ¼ 2 Dg x Dg x ¼ xv ; ¼ xu : 2 u 2 v
ð5:20Þ ð5:21Þ
6. For the surface x ¼ ðu; v ; uv ; 1ÞTran , Similar, from Eq. (2.2) we get
g 11 ¼ g 22 ¼ 0;
g 12 ¼ g 21 ¼ 1:
ð5:22Þ
Then we also get
Dg x ¼ ð0; 0; 1; 0ÞTran : 2
ð5:23Þ
So the structure equations can be give by
Dg x ; xuu ¼ 0; xuv ¼ 2 Dg x Dg x ¼ ¼ 0: 2 u 2 v 7. By taking kðuÞ 1; alent to
xvv ¼ 0;
ð5:24Þ ð5:25Þ
lðuÞ 1; c ¼ 1 in the last item of Theorem 3.1, we can get a surface which is centroaffinely equivTran
x ¼ v ð0; 2eu ; 0; 2eu Þ
Tran
þ ðeu ; ueu ; eu ; ueu Þ
:
ð5:26Þ
Again, we obtain the centroaffine metric of x from Eq. (2.2)
g 11 ¼ g 22 ¼ 0;
g 12 ¼ g 21 ¼ 4:
ð5:27Þ
A straightforward computation shows
Dg x 1 Tran ¼ ð0; eu ; 0; eu Þ : 2 2
ð5:28Þ
Y. Yang et al. / Applied Mathematics and Computation 243 (2014) 775–788
787
It is easy to deduce the structure equations of x, that is,
xuu ¼ xv þ x; Dg x ; xuv ¼ 4 2
ð5:29Þ
xvv ¼ 0; Dg x 1 ¼ xv ; 2 u 4 Dg x ¼ 0: 2 v
ð5:31Þ
ð5:30Þ
ð5:32Þ ð5:33Þ
Tran Tran 8. The surface x ¼ u; v ; u1v ; 1 is centroaffinely equivalent to x ¼ ðeuþv ; euv ; e2u ; 1Þ . From Eq. (2.2) we obtain its centroaffine metric
6
g 11 ¼
3
1 4
g 22 ¼
2 1
34
;
g 12 ¼ g 21 ¼ 0:
ð5:34Þ
The second transversal vector field
Tran 3 Dg x ¼ 34 euþv ; euv ; e2u ; 0 : 2
ð5:35Þ
Then the structure equations of x are
xuu ¼ xu þ g 11
Dg x ; 2
ð5:36Þ
xuv ¼ xv ; 1 Dg x ; xvv ¼ xu þ g 22 3 2 3 Dg x ¼ 34 xu ; 2 u 3 Dg x ¼ 3 4 xv : 2 v
ð5:37Þ ð5:38Þ ð5:39Þ ð5:40Þ
Tran 9. The surface x ¼ u2 ; cosu v ; sinu v ; 1 is centroaffinely equivalent to uþv
x¼
e
uþ2 v
;e
! ! !Tran pffiffiffi pffiffiffi 3 3 uþ2 v cos ðu v Þ ; e sin ðu v Þ ; 1 : 2 2
From Eq. (2.2), it is direct to get
g 11 ¼ g 22 ¼ 0;
g 12 ¼ g 21 ¼
14 pffiffiffi 27 3 3 : 4 2
ð5:41Þ
A straightforward computation shows
14 pffiffiffi 27 3 3 Dg x ¼ x ð0; 0; 0; 1ÞTran : 4 2 2
ð5:42Þ
So the structure equations of x are
xuu ¼ xv ;
ð5:43Þ
xuv ¼ x ð0; 0; 0; 1ÞTran ;
ð5:44Þ
xvv ¼ xu ; 14 pffiffiffi 27 3 3 Dg x ¼ xu ; 4 2 u 2 14 pffiffiffi 27 3 3 Dg x ¼ xv : 4 2 v 2
ð5:45Þ ð5:46Þ ð5:47Þ
10. For centroaffine umbilical surface x ¼ v V 1 ðuÞ þ V 2 ðuÞ, where V 001 ðuÞ ¼ kðuÞV 1 ðuÞ; V 002 ðuÞ ¼ lðuÞV 1 ðuÞ, and kðuÞ; arbitrary 1-variable functions, if we choose k ¼ 1; l ¼ sin u, the surface is centroaffinely equivalent to
x¼
v eu
eu cos u eu cos u ; v eu ; u; 1 2 2
Tran :
lðuÞ are
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Then we obtain
g 11 ¼ g 22 ¼ 0;
3
g 12 ¼ g 21 ¼ 24
ð5:48Þ
and
3 Dg x ; xuu ¼ ðv þ sin uÞxv ; xuv ¼ 24 2 3 3 Dg x Dg x ¼ xv ; 24 ¼ 0: 24 2 u 2 v
xvv ¼ 0;
ð5:49Þ ð5:50Þ
Acknowledgments The first author would express his gratitude to professor Lihe Wang for his hospitality and helpful discussions during the first author’s visits to Shanghai Jiao Tong University and University of Iowa. The authors would like to thank the referees for their comments which have improved the paper presentation. This work was supported by the Fundamental Research Funds for the Central Universities (No. N130405006) and NSFC (Nos. 11201056 and 11371080), and the first author was financial supported by the China Scholarship Council. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
W. Blaschke, Vorlesungen über Differentialgeometrie II, Affine Differentialgeometrie, Springer, Berlin, 1923. H. Furuhata, Minimal centroaffine immersions of codimension two, Bull. Belg. Math. Soc. 7 (2000) 125–134. A.M. Li, U. Simon, G. Zhao, Global Affine Differential Geometry of Hypersurfaces, W. De Gruyter, Berlin-New York, 1993. H.L. Liu, Classification of surfaces in R3 which are centroaffine-minimal and equiaffine-minimal, Bull. Belg. Math. Soc. 3 (1996) 577–583. H.L. Liu, Indefinite equi-centroaffinely homogeneous surfaces with vanishing Pick invariant in R4 , Hokkaido Math. J. 26 (1997) 225–251. H.L. Liu, Equi-centroaffinely homogeneous surfaces with vanishing Pick invariant in R4 , Proceeding of 1-st Non-Orthodox School on Nonlinearity & Geometry, 1998, pp. 335–340. Huili Liu, Seoung Dal Jung, Hypersurfaces which are equiaffine extremal and centroaffine extremal, Bull. Braz. Math. Soc. New Ser. 38 (4) (2007) 555– 571. K. Nomizu, T. Sasaki, Centroaffine immsersions of codimension two and projective hypersuface theory, Nagoya Math. J. 132 (1993) 63–90. K. Nomizu, T. Sasaki, Affine Differential Geometry, Cambridge University Press, 1994. R. Walter, Centroaffine differential geometry: submanifolds of codimension 2, Result Math. 13 (1988) 386–402. C.P. Wang, Centroaffine minimal hypersurfaces in Rnþ1 , Geom. Dedicata 51 (1994) 63–74. Y. Yang, H.L. Liu, Minimal centroaffine immersions of codimension two, Result Math. 52 (2008) 423–437. Y. Yang, Y.H. Yu, H.L. Liu, Centroaffine translation surfaces in R3 , Result Math. 56 (2009) 197–210. Y. Yang, Y.H. Yu, H.L. Liu, Linear Weingarten centroaffine translation surfaces in R3 , J. Math. Anal. Appl. 375 (2011) 458–466. Y. Yang, Y.H. Yu, H.L. Liu, Flat centroaffine surfaces with the degenerate second fundamental form and vanishing Pick invariant in R4 , J. Math. Anal. Appl. 397 (2013) 161–171. Y.H. Yu, Y. Yang, H.L. Liu, Centroaffine ruled surfaces in R3 , J. Math. Anal. Appl. 365 (2010) 683–693.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Flat centroaffine surfaces with parallel second fundamental form in R4 Yun Yang, Yanhua Yu ⇑, Huili Liu Department of Mathematics, Northeastern University, Shenyang, Liaoning 110004, PR China
a r t i c l e
i n f o
Keywords: Centroaffine surfaces Curvature Pick invariant The second fundamental form
a b s t r a c t By solving certain partial differential equations, we obtain some classification results for flat centroaffine surfaces in R4 , under the condition of parallel second fundamental form and vanishing Pick invariant. We also get some examples for flat centroaffine umbilical surface with parallel second fundamental form. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction The main purpose of affine differential geometry, as we know, is to study the properties of submanifolds M n of Rm that are invariant under the group of all affine transformations [9,3]. The classical theory for affine hypersurfaces was developed by Blaschke and his school [1], and has been reorganized in the last 20 years as geometry of affine immersions [8]. Similarly, the centroaffine differential geometry is the study of the properties of submanifolds that are invariant under the centroaffine transformation group, which is the subgroup of the affine transformation group that keeps the origin invariant. In centroaffine differential geometry, the theory of hypersurfaces has a long history. The notion of centroaffine minimal hypersurfaces was introduced by Wang [11] as extremals for the area integral of the centroaffine metric. The third author of this paper got the classification of surfaces in R3 which are both centroaffine-minimal and equiaffine-minimal [4], for hypersurfaces, see [7]. See also [13,14,16] for the classification results about centroaffine translation surfaces and centroaffine ruled surfaces in R3 . However, relatively little has been achieved in the study of centroaffine immersions with higher codimensions. For a centroaffine immersion into the affine space, the position vector yields its first canonical transversal vector field. A standard method of choosing a second one was proposed in 1950 by Lops˘ic (see Walter [10]). Reorganizing geometry of equi-centroaffine immersions of codimension two, Nomizu and Sasaki [8] took the prenormalized Blaschke normal field as the second canonical transversal vector field, and using this structure Furuhata [2] proved that an equi-centroaffine immersion is minimal if and only if the trace of the affine shape operator with respect to the prenormalized Blaschke normal field vanishes identically. On the other hand, the third author of this paper gave another structure to study the centroaffine immersions of codimension two [5,6]. He defined the centroaffine metric g of centroaffine immersions with codimension two in a new way and then chose Dg x as the second transversal vector field, where Dg denotes the Laplacian of g. According to this structure and using the efficacious moving frames method, the authors of this paper compared these different normalizations and defined the minimal centroaffine immersions of codimension two [12]. Recently, the authors of this paper also obtained some classification results for flat centroaffine surfaces in R4 with the condition of degenerate second fundamental form and vanishing Pick invariant [15]. In this paper, we further consider the centroaffine invariants of centroaffine
⇑ Corresponding author. E-mail addresses: [email protected] (Y. Yang), [email protected] (Y. Yu), [email protected] (H. Liu). http://dx.doi.org/10.1016/j.amc.2014.06.062 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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immersion x : M ¼ Mn ! Rnþ2 ðn P 2Þ, and find that the parallel property of the second fundamental form and umbilical property for centroaffine immersions of codimension two are invariant under centroaffine transformation. Then by solving certain systems of partial differential equations, we obtain a complete classification of flat centroaffine surfaces with parallel second fundamental form and vanishing Pick invariant in R4 . At the same time, we also get some examples for flat centroaffine umbilical surface with parallel second fundamental form. This paper is organized as follows. In Section 2 we recall the basic theory and notions for centroaffine immersions of codimension 2 with normalization fx; Dg xg. Note, there are some new definitions. In Section 3 we consider centroaffine surfaces with rh ¼ 0 in R4 , and then get a complete classification ~ h ¼ 0 in R4 , and also get a for flat surfaces with rh ¼ 0 and J ¼ 0. In Section 4 we talk about centroaffine surfaces with r ~ complete classification for flat surfaces with rh ¼ 0 and J ¼ 0. Furthermore, some flat centroaffine umbilical surfaces with parallel second fundamental form are obtained in Sections 3 and 4. Finally, in Section 5, we will give some more details for centroaffine surfaces with parallel second fundamental form appearing in Sections 3 and 4. 2. Centroaffine immersions in Rnþ2 We will recall the basic centroaffine theory in Rnþ2 . More details can be found in [12,15]. Let x : M ¼ M n ! Rnþ2 ðn P 2Þ be an oriented immersed submanifold such that xðpÞ R dxðT p MÞ for all p 2 M, and xðMÞ be not contained in a hyperplane containing the origin of Rnþ2 . Our convention for the range of indices is the following
1 6 i; j; k; . . . 6 n; n þ 1 6 a; b; c; . . . 6 n þ 2; 1 6 A; B; C; . . . 6 n þ 2 and we shall follow the usual Einstein summation convention. For any local oriented basis r ¼ fE1 ; E2 ; . . . ; En g of TM with dual basis fh1 ; h2 ; . . . ; hn g we define 2
G :¼ ½E1 ðxÞ; . . . ; En ðxÞ; x; d x ¼ Gij hi hj ;
ð2:1Þ nþ2
where ½ is the standard determinant in R that G is nondegenerate. We define
g :¼ g ij hi hj ;
, Gij :¼ ½E1 ðxÞ; . . . ; En ðxÞ; x; Ei Ej ðxÞ. G is a symmetric 2-form and we assume
1
g ij ¼ j det Gpq jnþ2 Gij :
ð2:2Þ
Then g is independent of the choice of the basis r and thus a globally defined symmetric 2-form. From Eq. (2.1) we know that the conformal class of g is a centroaffine invariant [5,6]. Definition 2.1. g is called the centroaffine metric of centroaffine immersion x : M ! Rnþ2 . Remark. x is called a nondegenerate centroaffine submanifold if g is nondegenerate and x is definite or indefinite if g is definite or indefinite, respectively. n o D x Definition 2.2. Let Dg denote the Laplacian of g. x; ng is called the centroaffine normalization of centroaffine immersion x : M ! Rnþ2 . Let U be a connected open subset of manifold M. If no confusion is possible, we will identify U with its image xðUÞ and T u M n o D x with x ðT u MÞ. We denote the canonical flat connection on Rnþ2 by D. With respect to the frame E1 ðxÞ; . . . ; En ðxÞ; x; ng , the structure equations for arbitrary tangential vector fields X; Y 2 XðUÞ can be written in the form [12]:
Dg x DY X ¼ rY X þ hðX; YÞx þ gðX; YÞ ; n Dg x ¼ SðXÞ þ qðXÞx: DX n
ð2:3Þ ð2:4Þ
A standard proof shows that r is a torsion free affine connection, h are symmetric bilinear forms, shape operator S is a (1, 1)-tensor field and q are 1-forms defined on U[12]. If we denote by Rr the curvature tensor of r, i.e., for arbitrary tangent vector fields X; Y; Z 2 XðUÞ; Rr ðX; YÞZ ¼ rX rY Z rY rX Z r½X;Y Z, then the centroaffine integrability conditions are
Rr ðX; YÞZ ¼ gðY; ZÞSðXÞ hðY; ZÞX gðX; ZÞSðYÞ þ hðX; ZÞY;
ð2:5Þ
ðrX gÞðY; ZÞ ¼ ðrY gÞðX; ZÞ;
ð2:6Þ
ðrX SÞðYÞ þ qðXÞY ¼ ðrY SÞðXÞ þ qðYÞX;
ð2:7Þ
hðX; SðYÞÞ hðY; SXÞ ¼ ðrX qÞðYÞ ðrY qÞðXÞ;
ð2:8Þ
gðX; SðYÞÞ gðY; SðXÞÞ ¼ 0:
ð2:9Þ
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~ ¼ fC ~ k g be the induced connection and the Levi–Civita connection of centroaffine In local coordinates, let r ¼ fCkij g and r ij metric g, and then we define the Fubini-Pick form C by
~k; C kij :¼ Ckij C ij C ijk :¼
ð2:10Þ
g il C ljk :
ð2:11Þ
The Pick invariant is defined by
J¼
1 g ij C lik C klj : nðn 1Þ
In [12] we obtained that C kij is symmetric for i and j and C ijk is symmetric for i; j and k. Furthermore we have
g ij C kij ¼ 0;
g ij hij ¼ 0;
C iij ¼ 0:
ð2:12Þ
~ij ¼ g Sl and H ~ij Þ, ~ ¼ ðh Introducing now h il j
H¼
~ trg H trS ¼ n n
ð2:13Þ
is called the centroaffine mean curvature of centroaffine immersion x : M n ! Rnþ2 , and the centroaffine immersion x is centroaffine minimal if and only if H ¼ 0 [12]. Certainly, we also have the centroaffine theorema egregium [15]
v¼
~ jk g jk R R ¼ ¼ J H; nðn 1Þ nðn 1Þ
ð2:14Þ
where R is the scalar curvature, v is the normalized scalar curvature of the metric g. Here we can give the following two definitions for centroaffine immersion of codimension two. Definition 2.3
L ¼ hij dxi dxj is call the second fundamental form of centroaffine immersion x : M ! Rnþ2 . Then centroaffine immersion x is said to have ~ h ¼ 0. parallel second fundamental form if rh ¼ 0 or r Definition 2.4. A surface in R4 is called centroaffine umbilical if the shape operator S has only one eigenvalue. : M ! Rnþ2 are centroaffine equivalent, then g ¼ kg, where k is constant, and it Remark. If two centroaffine immersion x; x ~ is easy to verify rh ¼ 0; rh ¼ 0 and umbilical property are invariant under centroaffine transformation from the structure Eqs. (2.3) and (2.4).
3. Flat centroaffine surfaces with $h ¼ 0 In this section we will deduce the following theorem which is a complete classification for flat centroaffine surfaces with J ¼ 0 and rh ¼ 0 in R4 . Theorem 3.1. Let x : M ! R4 be a flat centroaffine surface with J ¼ 0 and rh ¼ 0, then x is centroaffinely equivalent to one of the following surfaces in R4 : 1. 2. 3. 4. 5. 6.
Tran
x ¼ ðu2 þ v 2 ; u; v ; 1Þ ; Tran ðcosh u; sinh u; cos v ; sin v Þ ; Tran ðcosh u; sinh u; cosh v ; sinh v Þ ; Tran ðcos u; sin u; cos v ; sin v Þ ; Tran ðcosh u cos v ; cosh u sin v ; sinh u cos v ; sinh u sin v Þ ; x ¼ v V 1 ðuÞ þ V 2 ðuÞ where V 1 ðuÞ; V 2 ðuÞ are linearly independent 1-variable vector kðuÞ lðuÞ ; kðuÞ and lðuÞ are arbitrary 1-variable functions. ðV 001 ðuÞ; V 002 ðuÞÞ ¼ ðV 1 ðuÞ; V 2 ðuÞÞ 0 c
fields
satisfying
that
Remark. From the centroaffine theorema egregium (2.14), we know flat centroaffine surfaces with vanishing Pick invariant are centroaffine minimal, so the surfaces included in Theorem 3.1 all are centroaffine minimal.
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Especially, in this section, we also find the following examples of flat centroaffine umbilical surfaces with rh ¼ 0: Tran
ðu2 þ v 2 ; u; v ; 1Þ ; Tran 1 u; v ; ;1 ; uv Tran 1 1 u2 ; sin v ; cos v ; 1 ; u u
ð3:1Þ ð3:2Þ ð3:3Þ
v V 1 ðuÞ þ V 2 ðuÞ;
ð3:4Þ V 001 ðuÞ
V 002 ðuÞ
where V 1 ðuÞ; V 2 ðuÞ are linearly independent 1-variable vector fields satisfying that ¼ kðuÞV 1 ðuÞ; ¼ lðuÞV 1 ðuÞ, and kðuÞ; lðuÞ are arbitrary 1-variable functions. Exactly, in following Section 3.1 we will get the classifications of definite centroaffine surfaces with J ¼ 0 and rh ¼ 0. Then under the same condition, the classifications of indefinite centroaffine surfaces will be obtained in Section 3.2. At the same time, the umbilical surfaces will be considered in Sections 3.1 and 3.2. 3.1. Definite centroaffine surfaces in R4 Let x : M ! R4 be a centroaffine surface with definite centroaffine metric g. Hence we can choose a local basis r ¼ fE1 ; E2 g of TM such that
g ¼ ex ðdu2 þ dv 2 Þ;
ð3:5Þ
where ¼ 1. Obviously, g 11 ¼ g 22 ¼ From Eq. (2.12) we get
ex ;
4
g 12 ¼ g 21 ¼ 0, and fE1 ðxÞ; E2 ðxÞ; x; Dg xg forms a local moving frame for R along M.
g ij hij ¼ ex ðh11 þ h22 Þ ¼ 0;
ð3:6Þ
that is
h11 þ h22 ¼ 0:
ð3:7Þ
By a direct calculation, we know that the coefficients of the Levi–Civita connection of the centroaffine metric g are
~1 ¼ C 11 ~1 ¼ C 12
xu 2
;
~ 2 ¼ xv ; C 11
;
~2 ¼ C 12
xv 2
xu
~1 ¼ C 22
2
xu
ð3:9Þ
;
2
~2 ¼ C 22
;
ð3:8Þ
2
xv 2
ð3:10Þ
:
Again from Eq. (2.12) it is easy to find
C111 þ C212 ¼ xu ;
C112 þ C222 ¼ xv ;
C111 þ C122 ¼ 0;
C211 þ C222 ¼ 0:
ð3:11Þ
Then the Pick invariant
J¼
ex 2
2
2
ððxu 2C111 Þ þ ðxv 2C222 Þ Þ:
ð3:12Þ
By the flatness assumption of x we can choose the coordinates such that x 0, which leads to
C111 ¼ C212 ¼ C122 ¼ aðu; v Þ; 2 22
C
¼ C 2
1 12
¼ C
2 11
¼ bðu; v Þ;
2
J ¼ 2ða þ b Þ;
ð3:13Þ ð3:14Þ ð3:15Þ
where the notations a; b are mainly for simplification. According to the assumption rh ¼ 0 and Eq. (3.7), a direct calculation yields
@h11 ¼ 2ðah11 bh12 Þ; @u @h11 ¼ 2ðah11 bh12 Þ; @u @h11 ¼ 2ðbh11 ah12 Þ; @v @h11 ¼ 2ðbh11 ah12 Þ; @v
ð3:16Þ ð3:17Þ ð3:18Þ ð3:19Þ
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@h12 ¼ 0; @u @h12 ¼ 0: @v
ð3:20Þ ð3:21Þ
It immediately follows that h12 is constant and
a b b
a
h11 h12
¼ 0:
ð3:22Þ 2
Surely, the direct result of Eq. (3.22) is a2 þ b ¼ 0 or h11 ¼ h12 ¼ 0. In the following we consider them respectively. 2
Case 1. a2 þ b ¼ 0. 2 According to Eq. (3.15), it is clear that a2 þ b ¼ 0 is equivalent to J ¼ 0, and also a ¼ b ¼ 0. Again from Eqs. (3.16)–(3.21), we can find hij are constants, where i; j 2 f1; 2g. Therefore, the structure equations can be changed as
xuu ¼ h11 x þ
Dg x ; 2
ð3:23Þ
xuv ¼ h12 x;
ð3:24Þ
Dg x ; xvv ¼ h22 x þ 2 D x g ¼ h22 xu þ h12 xv ; 2 u D x g ¼ h12 xu h11 xv : 2 v
ð3:25Þ ð3:26Þ ð3:27Þ
If h12 ¼ 0, we can get a system of partial differential equations from Eqs. (3.23) and (3.24),
xuv ¼ 0;
ð3:28Þ
xuu xvv ¼ ðh11 h22 Þx:
ð3:29Þ
The condition h11 h22 ¼ 0 implies the surface is centroaffinely equivalent to [15] Tran
ðu2 þ v 2 ; u; v ; 1Þ
ð3:30Þ
:
On the other hand, the condition h11 h22 – 0 deduces the surface is centroaffinely equivalent to [6] Tran
ðcosh u; sinh u; cos v ; sin v Þ
:
ð3:31Þ
Obviously, these two surfaces are (1) and (2) in Theorem 3.1. Tran
Remark. Eq. (3.7) and h11 h22 ¼ 0 can lead to h11 ¼ h22 ¼ 0, so it is easy to see the surface ðu2 þ v 2 ; u; v ; 1Þ is a centroaffine umbilical surface with shape operator S ¼ 0. If h12 – 0 and h11 h22 ¼ 0, which implies h11 ¼ h22 ¼ 0 from Eq. (3.7), then parameter transformation ¼ u þ v; v ¼ u v changes the structure Eqs. (3.23)–(3.25) to u
xuu xv v ¼ h12 x;
ð3:32Þ
xuv ¼ 0:
ð3:33Þ
This surface, indeed, is centroaffinely equivalent to (3.31). If h12 – 0 and h11 h22 – 0, which means h11 ¼ h22 – 0 from Eq. (3.7), we can take
0
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 h h11 A 11 ¼@ 1þ u þ v; u h12 h12 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 h11 h11 A @ v ¼ þ 1þ u þ v: h12 h12
ð3:34Þ
ð3:35Þ
Then Eqs. (3.23)–(3.25), that is, the system
xuv ¼ h12 x;
ð3:36Þ
xuu xvv ¼ 2h11 x;
ð3:37Þ
xuu xvv 2
h11 xu v ¼ 0 h12
ð3:38Þ
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can be changed to
xuv ¼ 0; 0 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 2 2 h h h h11 A 11 11 11 @ Axuu þ @ xv v ¼ h12 x: 1þ þ 1þ h12 h12 h12 h12
ð3:39Þ ð3:40Þ
By solving this system of partial differential equations, we know this surface is also centroaffinely equivalent to (3.31). 2
Case 2. a2 þ b – 0. 2 From Eq. (3.22), we know a2 þ b – 0 implies h11 ¼ h12 ¼ 0. Together with Eq. (3.7), it is direct to see hij ¼ 0, where i; j 2 f1; 2g. Then the structure equations can be rewritten as
xuu ¼ axu bxv þ
Dg x ; 2
ð3:41Þ
xuv ¼ bxu axv ;
ð3:42Þ
Dg x xvv ¼ axu þ bxv þ ; 2 D x 2 g ¼ ð2a2 þ 2b þ au bv Þxu þ ðav bu Þxv ; 2 u D x 2 g ¼ ðav bu Þxu þ ð2a2 þ 2b þ bv au Þxv : 2 v
ð3:43Þ ð3:44Þ ð3:45Þ
Here we assume x is a centroaffine umbilical surface, which implies the shape operator S has only one eigenvalue kðu; v Þ. From Eqs. (3.44) and (3.45), we have
ðSij Þ
¼
ð2a2 þ 2b þ au bv Þ
2
av þ bu
av þ bu
ð2a2 þ 2b þ bv au Þ
2
! :
ð3:46Þ
It immediately follows that
8 > < au ¼ bv ; av ¼ bu ; > : 2 2 2a þ 2b ¼ k: Then by
with u and
Dg x 2 uv
au av
¼
Dg x 2
vu
ð3:47Þ we get ku ¼ kv ¼ 0, i.e., k is nonzero constant. Differentiating both sides of the above last equation
v respectively, we have bu bv
a b
¼ 0:
ð3:48Þ
2
Together with a2 þ b – 0, the first two equations of (3.47) generate
a2u þ a2v ¼ 0: It immediately deduces a and b are constants. Then from Eqs. (3.41) and (3.42), we obtain
ðeauþbv xÞuv ¼ abðeauþbv xÞ;
ð3:49Þ 2
ðeðauþbv Þ xÞuu ðeðauþbv Þ xÞvv ¼ ða2 b Þðeðauþbv Þ xÞ:
ð3:50Þ
Remark. Obviously, if x is a solution of Eqs. (3.49) and (3.50), so is x þ V, where V is an arbitrary constant vector. In the following, we will give a solution for Eqs. (3.49) and (3.50) get a flat centroaffine umbilical surface with parallel second fundamental form. (1) If ab ¼ 0, then x ¼ eðauþbv Þ ðV 1 ðuÞ þ V 2 ðv ÞÞ, where V 1 ðuÞ; V 2 ðv Þ are 1-variable vector fields with respect to u and v respectively. Without loss of generality, we may assume that a ¼ 0 and x ¼ ebv ðV 1 ðuÞ þ V 2 ðv ÞÞ. Combining of Eqs. (3.41) and (3.43) leads to xuu xvv ¼ 2bxv , so it follows that
V 1 ðuÞ ¼ e
pffiffi 3bu bv
3 þ e V 2 ðv Þ ¼ e V
3bv
pffiffi 3bu
V 2 V 0 ; V 4 þ V 0 ;
V 1 þ e
ð3:51Þ ð3:52Þ
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Y. Yang et al. / Applied Mathematics and Computation 243 (2014) 775–788
i ; i 2 f0; 1; 2; 3; 4g are constant vectors and V 1 ; V 2 ; V 3 ; V 4 are linearly independent. Thus the surface is centroaffwhere V inely equivalent to
ðeuþv ; euv ; e2u ; 1Þ
Tran
ð3:53Þ
:
By a parameter transformation, this surface is equivalent to
Tran 1 u; v ; ;1 ; uv
ð3:54Þ
which is an example of flat centroaffine umbilical surfaces with parallel second fundamental form. 2 ¼ au þ bv ; v ¼ au bv , we can obtain the same surface as Eq. (2) If ab – 0 and a2 ¼ b , by using variable substitution u (3.53). 3.2. Indefinite centroaffine surfaces in R4 Let x : M ! R4 be a centroaffine surface with indefinite centroaffine metric g. By choosing the asymptotic local basis
r ¼ fE1 ; E2 g of TM such that g ¼ 2ex ðdudv Þ ¼ ex du dv þ ex dv du; we get g 11 ¼ g 22 ¼ 0; g 12 ¼ g 21 ¼ Eq. (2.12) we get
ex ,
ð3:55Þ 4
and fE1 ðxÞ; E2 ðxÞ; x; Dg xg forms a local moving frame for R along M. Again from
g ij hij ¼ ew ðh12 þ h21 Þ ¼ 0;
ð3:56Þ
that is,
h12 þ h21 ¼ 2h12 ¼ 0:
ð3:57Þ
The structure equations can be rewritten as
xuu ¼ C111 xu þ C211 xv þ h11 x; Dg x xuv ¼ C112 xu þ C212 xv þ ex ; 2
ð3:58Þ
xvv ¼ C122 xu þ C222 xv þ h22 x; Dg x ¼ S11 xu S21 xv þ q1 x; 2 u Dg x ¼ S12 xu S22 xv þ q2 x: 2 v
ð3:60Þ
ð3:59Þ
ð3:61Þ ð3:62Þ
By a direct calculation, we can get the coefficients of the Levi–Civita connection of the centroaffine metric g
~ 1 ¼ xu ; C 11 ~ 1 ¼ 0; C 12 ~ 1 ¼ 0; C 22
~ 2 ¼ 0; C 11 2 ~ C12 ¼ 0; ~ 2 ¼ xv : C
ð3:63Þ ð3:64Þ ð3:65Þ
22
According to Eq. (2.12) we have
C111 ¼ xu ;
C112 ¼ C212 ¼ 0;
C222 ¼ xv :
ð3:66Þ 4
Similarly, we get the Pick invariant of indefinite centroaffine surface x in R
J ¼ ex C211 C122 :
ð3:67Þ
Then by the assumption of flatness for x, we can choose the coordinates such that x 0. Thus combing of Eqs. (3.66) and (3.67) gives
C111 ¼ C112 ¼ C212 ¼ C222 ¼ 0:
ð3:68Þ
The assumption rh ¼ 0, together with Eq. (3.57), yields
@h11 ¼ 0; @u @h22 ¼ 0; @v
@h11 ¼ 0; @v
C122 h11 ¼ 0;
@h22 ¼ 0; @u
C211 h22 ¼ 0:
ð3:69Þ ð3:70Þ
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It immediately follows that h11 and h22 are constants. Depending on J ¼ 0 and J – 0, in the following we will divide into two cases. Case 1. C211 C122 ¼ 0. From Eq. (3.67), it is clear that C211 C122 ¼ 0 is equivalent to J ¼ 0. According to the different situations of hij ; i; j 2 f1; 2g, we will have the following classifications. (1)
If h11 h22 – 0, Eq. (3.70) leads to C211 ¼ C122 ¼ 0. According to Eqs. (3.58)–(3.62), we have
xuu ¼ h11 x;
xu v ¼
Dg x ¼ h11 xv ; 2 u
Dg x ; xvv ¼ h22 x; 2 Dg x ¼ h22 xu : 2 v
ð3:71Þ
ð3:72Þ
Firstly, if h11 > 0 and h22 > 0, by solving the system of partial equation (3.71) we have
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi x ¼ e h11 uþ h22 v V 1 þ e h11 u h22 v V 2 þ e h11 u h22 v V 3 þ e h11 uþ h22 v V 4 ; where V i ; i 2 f1; 2; 3; 4g are four linearly independent constant vectors, and this surface is centroaffinely equivalent to
ðcosh u; sinh u; cosh v ; sinh v Þ
Tran
ð3:73Þ
:
Then if h11 < 0 and h22 < 0, by solving the system of partial equation (3.71) we have
x ¼ cos u cos v V 1 þ cos u sin v V 2 þ sin u cos v V 3 þ sin u sin v V 4 ; where V i ; i 2 f1; 2; 3; 4g are four linearly independent constant vectors, and this surface is centroaffinely equivalent to
ðcos u; sin u; cos v ; sin v Þ
Tran
ð3:74Þ
:
Otherwise, if h11 h22 < 0, by solving the system of partial equation (3.71), the surface is centroaffinely equivalent to
ðcosh u cos v ; cosh u sin v ; sinh u cos v ; sinh u sin v Þ
Tran
:
ð3:75Þ
These three surfaces are surfaces (3), (4) and (5) in Theorem 3.1 respectively. (2) If h11 ¼ h22 ¼ 0 and C211 ¼ C122 ¼ 0, the structure Eqs. (3.58)–(3.61), can be rewritten as
Dg x xuu ¼ xvv ¼ 0; xu v ¼ ; 2 Dg x Dg x ¼ ¼ 0: 2 u 2 v
ð3:76Þ ð3:77Þ
By solving this system of partial differential equations, this surface is centroaffinely equivalent to
ðu; v ; uv ; 1ÞTran ;
ð3:78Þ
which is included in the surface (6) of Theorem 3.1. Obviously, its shape operator S ¼ 0, so this surface is one example of flat centroaffine umbilical surfaces with rh ¼ 0. (3) If h11 ¼ h22 ¼ 0 and one of C211 and C122 is not zero, without loss of generality we can assume C211 – 0. Eqs. (3.58) and (3.62) can be rewritten as
xuu ¼ C211 xv ;
xu v ¼
Dg x @ C211 ¼ xv ; 2 u @v
Dg x ; 2
xvv ¼ 0;
ð3:79Þ
Dg x ¼ 0: 2 v
ð3:80Þ @ 2 C2
Then we can get the integrability conditions @ v 211 ¼ 0, which implies C211 ¼ kðuÞv þ lðuÞ, where kðuÞ; lðuÞ are arbitrary 1-variable functions and k2 ðuÞ þ l2 ðuÞ – 0. Therefore, the surface is centroaffinely equivalent to
v V 1 ðuÞ þ V 2 ðuÞ; where V 1 ðuÞ; V 2 ðuÞ are linearly independent 1-variable vector fields satisfying that V 001 ðuÞ ¼ kðuÞV 1 ðuÞ; V 002 ðuÞ ¼ lðuÞV 1 ðuÞ. Surely, this surfaces is also included in the surface (6) of Theorem 3.1. From Eq. (3.80) we know that this surface is also one of flat centroaffine umbilical surfaces. (4) If one of h11 and h22 is not zero, without loss of generality we can assume h22 ¼ 0 and h11 – 0. From Eq. (3.70) we obtain C122 ¼ 0. Eqs. (3.58) and (3.60) can be rewritten as
xuu ¼ C211 xv þ h11 x;
xvv ¼ 0:
ð3:81Þ
Y. Yang et al. / Applied Mathematics and Computation 243 (2014) 775–788
783
@ 2 C2
Using xuuvv ¼ xvv uu , we also get the integrability conditions @ v 211 ¼ 0, which implies C211 ¼ c1 ðuÞv þ c2 ðuÞ, where c1 ðuÞ; c2 ðuÞ are arbitrary 1-variable functions. Therefore, the surface is centroaffinely equivalent to
v V 1 ðuÞ þ V 2 ðuÞ; where V 1 ðuÞ; V 2 ðuÞ are linearly independent 1-variable vector fields satisfying that
ðV 001 ðuÞ; V 002 ðuÞÞ ¼ ðV 1 ðuÞ; V 2 ðuÞÞ
c1 ðuÞ þ h11
c2 ðuÞ
0
h11
:
Exactly, this surface is also included in (6) in Theorem 3.1. Case 2. C211 C122 – 0. From Eq. (3.70), it is clear that C211 C122 – 0 implies h11 ¼ h22 ¼ 0. Then the structure equations can be rewritten as
xuu ¼ C211 xv ; Dg x xuv ¼ ; 2 1 xvv ¼ C22 xu ; Dg x @ C2 ¼ C211 C122 xu þ 11 xv ; 2 u @v Dg x @ C122 2 ¼ xu þ C11 C122 xv : 2 v @u
ð3:82Þ ð3:83Þ ð3:84Þ ð3:85Þ ð3:86Þ
Here we also assume x is a centroaffine umbilical surface, that is, the shape operator S has only one eigenvalue kðu; v Þ. By a simple calculation, we obtain
C211 C122 ¼ k;
ðC211 Þv ðC122 Þu ¼ 0:
ð3:87Þ
The integrability condition generates
@ 2 C211 @ C122 2 @k ¼ C þ ; @v 2 @u 11 @u @ 2 C122 @ C211 1 @k ¼ C þ : @u2 @ v 22 @ v
ð3:88Þ ð3:89Þ
Especially, if k is nonzero constant, from Eqs. (3.87) and (3.89), we can deduce C211 and C122 are constants. Taking the notation a :¼ C211 ; b :¼ C122 for simplification, we obtain
xuu ¼ axv ;
xvv ¼ bxu :
ð3:90Þ
Due to a; b are constants, we have
xuuv ¼ axvv ¼ abxu ;
xvv u ¼ bxuu ¼ abxv ;
ð3:91Þ
which implies
ðxuv abxÞu ¼ 0;
ðxv u abxÞv ¼ 0:
ð3:92Þ
Therefore, we get
xuv abx ¼ abV 1 ;
ð3:93Þ
where V 1 is arbitrary constant vector. Together with Eq. (3.90), it follows that
ðx V 1 Þuu ¼ aðx V 1 Þv ;
ðx V 1 Þuv ¼ abðx V 1 Þ;
ðx V 1 Þvv ¼ bðx V 1 Þu :
ð3:94Þ
Assume a ¼ b. Obviously,
x ¼ V1 þ e
aðuþv Þ
V2 þ e
2aðuþv Þ
! ! pffiffiffi pffiffiffi 3a 3a 2aðuþv Þ cos ðu v Þ V 3 þ e sin ðu v Þ V 4 2 2
ð3:95Þ
gives a solution of Eq. (3.94), where V 1 ; V 2 ; V 3 ; V 4 are four linearly independent constant vectors. By parameter transformation and centroaffine transformation, this surface is equivalent to
Tran 1 1 u2 ; sin v ; cos v ; 1 ; u u which also is an example of flat centroaffine umbilical surfaces with parallel second fundamental form.
ð3:96Þ
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~ ¼0 4. Flat centroaffine surfaces with $h ~ h ¼ 0 implies all hij are constants, where i; j 2 f1; 2g. From the structure equaSince the centroaffine surface x is flat, r ~ h ¼ 0. tions in Section 3 it is not difficult to verify the surfaces (3.1)–(3.4) are also flat centroaffine umbilical surfaces with r For definite centroaffine surfaces with J ¼ 0, we can obtain its structure equations
xuu ¼ h11 x þ
Dg x ; 2
ð4:1Þ
xuv ¼ h12 x;
ð4:2Þ
Dg x xvv ¼ h11 x þ ; 2 D x g ¼ h12 xu h11 xv ; 2 u D x g ¼ h22 xu þ h12 xv : 2 v
ð4:3Þ ð4:4Þ ð4:5Þ
Then we get equations
xuv ¼ h12 x;
ð4:6Þ
xuu xvv ¼ 2h11 x:
ð4:7Þ
According to the calculations in Case 1 of Section 3.1, we know the surface is centroaffinely equivalent to Tran
ðcosh u; sinh u; cos v ; sin v Þ
ð4:8Þ
or
ðu2 þ v 2 ; u; v ; 1Þ
Tran
ð4:9Þ
:
For indefinite centroaffine surfaces with J ¼ 0, we get its structure equations
xuu ¼ C211 xv þ h11 x; xuv ¼
ð4:10Þ
Dg x ; 2
ð4:11Þ
xvv ¼ C122 xu þ h22 x; ! Dg x @ C211 ¼ þ h11 xv þ h22 C211 x; 2 u @v ! Dg x @ C122 ¼ þ h22 xu þ h11 C122 x; 2 u @u
ð4:12Þ ð4:13Þ
ð4:14Þ
where C122 C211 ¼ 0. Without loss of generality, we may assume that C122 ¼ 0. The integrability condition gives @ 2 C211 @v 2
@ C2
¼ 0; @v11 h22 ¼ 0. If h22 ¼ 0, the surface is centroaffinely equivalent to
v V 1 ðuÞ þ V 2 ðuÞ; where V 1 ðuÞ; V 2 ðuÞ are linearly independent 1-variable vector fields satisfying that
kðuÞ ðV 001 ðuÞ; V 002 ðuÞÞ ¼ ðV 1 ðuÞ; V 2 ðuÞÞ 0
lðuÞ c
; @ C2
7kðuÞ; lðuÞ are arbitrary 1-variable functions, and c is constant. If h22 – 0, then @v11 ¼ 0 and we may assume C211 ¼ f ðuÞ, where f ðuÞ is an arbitrary 1-variable function. Then if h22 > 0, the surface is centroaffinely equivalent to
pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi sinhð h22 v ÞV 1 ðuÞ þ coshð h22 v ÞV 2 ðuÞ; where V 1 ðuÞ; V 2 ðuÞ are linearly independent 1-variable vector fields satisfying that V 001 ðuÞ ¼ ðh11 þ pffiffiffiffiffiffiffi V 002 ðuÞ ¼ ðh11 h22 f ðuÞÞV 2 ðuÞ. Otherwise, if h22 < 0, the surface is centroaffinely equivalent to
sin
pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi h22 v V 1 ðuÞ þ cos h22 v V 2 ðuÞ;
pffiffiffiffiffiffiffi h22 f ðuÞÞV 1 ðuÞ and
Y. Yang et al. / Applied Mathematics and Computation 243 (2014) 775–788
785
where V 1 ðuÞ; V 2 ðuÞ are linearly independent 1-variable vector fields satisfying that
ðV 001 ðuÞ; V 002 ðuÞÞ ¼ ðV 1 ðuÞ; V 2 ðuÞÞ
! pffiffiffiffiffiffiffiffiffiffiffi h22 f ðuÞ
h11 pffiffiffiffiffiffiffiffiffiffiffi h22 f ðuÞ
h11
:
Therefore, we obtain the following theorem. ~ h ¼ 0, then x is centroaffinely equivalent to one of the Theorem 4.1. Let x : M ! R4 be a flat centroaffine surface with J ¼ 0 and r following surfaces in R4 : Tran
1. x ¼ ðu2 þ v 2 ; u; v ; 1Þ ; Tran 2. ðcosh u; sinh u; cos v ; sin v Þ ; 3. x ¼ v V 1 ðuÞ þ V 2 ðuÞ, where V 1 ðuÞ; V 2 ðuÞ are linearly independent 1-variable vector fields satisfying kðuÞ lðuÞ ; kðuÞ and lðuÞ are arbitrary 1-variable function, and c is constant; ðV 001 ðuÞ; V 002 ðuÞÞ ¼ ðV 1 ðuÞ; V 2 ðuÞÞ 0 c
that
4. x ¼ sinh v V 1 ðuÞ þ cosh v V 2 ðuÞ, where V 1 ðuÞ; V 2 ðuÞ are linearly independent 1-variable vector fields satisfying that V 001 ðuÞ ¼ ðc þ kðuÞÞV 1 ðuÞ and V 002 ðuÞ ¼ ðc kðuÞÞV 2 ðuÞ; kðuÞ is an arbitrary 1-variable functions, and c is constant; 5. sin v V 1 ðuÞ þ cos v V 2 ðuÞ, ðV 001 ðuÞ; V 002 ðuÞÞ
where V 1 ðuÞ; V 2 ðuÞ are linearly independent 1-variable vector fields satisfying c kðuÞ ; kðuÞ is an arbitrary 1-variable function, and c is nonzero constant. ¼ ðV 1 ðuÞ; V 2 ðuÞÞ kðuÞ c
that
Remark. From the centroaffine theorema egregium (2.14), we know flat centroaffine surfaces with vanishing Pick invariant are centroaffine minimal, so the surfaces included in Theorem 4.1 are centroaffine minimal. On the other hand, it is easy to find that all surfaces of Theorem 3.1 are some examples of Theorem 4.1. 5. The examples in Sections 3 and 4 In this section we will calculate the basic invariants for the surfaces appearing in Sections 3 and 4, and give their structure equations. Tran
1. For the surface x ¼ ðu2 þ v 2 ; u; v ; 1Þ
g 11 ¼ g 22
pffiffiffi ¼ 2;
, it follows from Eq. (2.2) that
g 12 ¼ g 21 ¼ 0:
ð5:1Þ
By a straightforward computation we get
pffiffiffi Tran Dg x ¼ ð 2; 0; 0; 0Þ : 2
ð5:2Þ
Thus, the structure equations can be written as
pffiffiffiDg x ; xuu ¼ xvv ¼ 2 2
ð5:3Þ
xuv ¼ 0; Dg x Dg x ¼ ¼ 0: 2 u 2 v 2. For the surface x ¼ ðcosh u; sinh u; cos v ; sin v Þ
g 11 ¼ g 22 ¼ 1;
ð5:4Þ ð5:5Þ Tran
, from Eq. (2.2) we can get the centroaffine metric
g 12 ¼ g 21 ¼ 0:
ð5:6Þ
Then we can obtain the second transversal vector field
Dg x 1 Tran ¼ ðcosh u; sinh u; cos v ; sin v Þ : 2 2
ð5:7Þ
Therefore, the structure equations are given by
1 Dg x 1 Dg x xuu ¼ x þ ; xuv ¼ 0; xvv ¼ x þ ; 2 2 2 2 Dg x 1 Dg x 1 ¼ xu ; ¼ xv : 2 u 2 2 v 2
ð5:8Þ ð5:9Þ Tran
3. For the surface x ¼ ðcosh u; sinh u; cosh v ; sinh v Þ
g 11 ¼ 1;
g 22 ¼ 1;
g 12 ¼ g 21 ¼ 0:
, from Eq. (2.2) we also get its centroaffine metric
ð5:10Þ
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Y. Yang et al. / Applied Mathematics and Computation 243 (2014) 775–788
A direct computation gives
Dg x 1 Tran ¼ ðcosh u; sinh u; cosh v ; sinh v Þ : 2 2
ð5:11Þ
Then, we have
1 Dg x 1 Dg x xuu ¼ x þ ; xuv ¼ 0; xvv ¼ x ; 2 2 2 2 Dg x 1 Dg x 1 ¼ xu ; ¼ xv : 2 u 2 2 v 2 4. For the surface x ¼ ðcos u; sin u; cos v ; sin v Þ
g 11 ¼ 1;
g 22 ¼ 1;
ð5:12Þ ð5:13Þ
Tran
, again from Eq. (2.2) it follows that
g 12 ¼ g 21 ¼ 0:
ð5:14Þ
By a straightforward computation we have
Dg x 1 Tran ¼ ðcos u; sin u; cos v ; sin v Þ : 2 2
ð5:15Þ
Then, the structure equations can be written as
1 Dg x 1 Dg x xuu ¼ x ; xuv ¼ 0; xvv ¼ x þ ; 2 2 2 2 Dg x 1 Dg x 1 ¼ xu ; ¼ xv : 2 u 2 2 v 2 5. For the surface x ¼ ðcosh u cos v ; cosh u sin v ; sinh u cos v ; cosh u sin v Þ
g 11 ¼ g 22 ¼ 0;
g 12 ¼ g 21 ¼ 1:
ð5:16Þ ð5:17Þ Tran
, it is easy to verify from Eq. (2.2) that
ð5:18Þ
A straightforward computation yields
Dg x Tran ¼ ðsinh u sin v ; sinh u cos v ; cosh u sin v ; cosh u cos v Þ : 2
ð5:19Þ
Thus, the structure equations can be represented as
Dg x ; xvv ¼ x; xuu ¼ x; xuv ¼ 2 Dg x Dg x ¼ xv ; ¼ xu : 2 u 2 v
ð5:20Þ ð5:21Þ
6. For the surface x ¼ ðu; v ; uv ; 1ÞTran , Similar, from Eq. (2.2) we get
g 11 ¼ g 22 ¼ 0;
g 12 ¼ g 21 ¼ 1:
ð5:22Þ
Then we also get
Dg x ¼ ð0; 0; 1; 0ÞTran : 2
ð5:23Þ
So the structure equations can be give by
Dg x ; xuu ¼ 0; xuv ¼ 2 Dg x Dg x ¼ ¼ 0: 2 u 2 v 7. By taking kðuÞ 1; alent to
xvv ¼ 0;
ð5:24Þ ð5:25Þ
lðuÞ 1; c ¼ 1 in the last item of Theorem 3.1, we can get a surface which is centroaffinely equivTran
x ¼ v ð0; 2eu ; 0; 2eu Þ
Tran
þ ðeu ; ueu ; eu ; ueu Þ
:
ð5:26Þ
Again, we obtain the centroaffine metric of x from Eq. (2.2)
g 11 ¼ g 22 ¼ 0;
g 12 ¼ g 21 ¼ 4:
ð5:27Þ
A straightforward computation shows
Dg x 1 Tran ¼ ð0; eu ; 0; eu Þ : 2 2
ð5:28Þ
Y. Yang et al. / Applied Mathematics and Computation 243 (2014) 775–788
787
It is easy to deduce the structure equations of x, that is,
xuu ¼ xv þ x; Dg x ; xuv ¼ 4 2
ð5:29Þ
xvv ¼ 0; Dg x 1 ¼ xv ; 2 u 4 Dg x ¼ 0: 2 v
ð5:31Þ
ð5:30Þ
ð5:32Þ ð5:33Þ
Tran Tran 8. The surface x ¼ u; v ; u1v ; 1 is centroaffinely equivalent to x ¼ ðeuþv ; euv ; e2u ; 1Þ . From Eq. (2.2) we obtain its centroaffine metric
6
g 11 ¼
3
1 4
g 22 ¼
2 1
34
;
g 12 ¼ g 21 ¼ 0:
ð5:34Þ
The second transversal vector field
Tran 3 Dg x ¼ 34 euþv ; euv ; e2u ; 0 : 2
ð5:35Þ
Then the structure equations of x are
xuu ¼ xu þ g 11
Dg x ; 2
ð5:36Þ
xuv ¼ xv ; 1 Dg x ; xvv ¼ xu þ g 22 3 2 3 Dg x ¼ 34 xu ; 2 u 3 Dg x ¼ 3 4 xv : 2 v
ð5:37Þ ð5:38Þ ð5:39Þ ð5:40Þ
Tran 9. The surface x ¼ u2 ; cosu v ; sinu v ; 1 is centroaffinely equivalent to uþv
x¼
e
uþ2 v
;e
! ! !Tran pffiffiffi pffiffiffi 3 3 uþ2 v cos ðu v Þ ; e sin ðu v Þ ; 1 : 2 2
From Eq. (2.2), it is direct to get
g 11 ¼ g 22 ¼ 0;
g 12 ¼ g 21 ¼
14 pffiffiffi 27 3 3 : 4 2
ð5:41Þ
A straightforward computation shows
14 pffiffiffi 27 3 3 Dg x ¼ x ð0; 0; 0; 1ÞTran : 4 2 2
ð5:42Þ
So the structure equations of x are
xuu ¼ xv ;
ð5:43Þ
xuv ¼ x ð0; 0; 0; 1ÞTran ;
ð5:44Þ
xvv ¼ xu ; 14 pffiffiffi 27 3 3 Dg x ¼ xu ; 4 2 u 2 14 pffiffiffi 27 3 3 Dg x ¼ xv : 4 2 v 2
ð5:45Þ ð5:46Þ ð5:47Þ
10. For centroaffine umbilical surface x ¼ v V 1 ðuÞ þ V 2 ðuÞ, where V 001 ðuÞ ¼ kðuÞV 1 ðuÞ; V 002 ðuÞ ¼ lðuÞV 1 ðuÞ, and kðuÞ; arbitrary 1-variable functions, if we choose k ¼ 1; l ¼ sin u, the surface is centroaffinely equivalent to
x¼
v eu
eu cos u eu cos u ; v eu ; u; 1 2 2
Tran :
lðuÞ are
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Y. Yang et al. / Applied Mathematics and Computation 243 (2014) 775–788
Then we obtain
g 11 ¼ g 22 ¼ 0;
3
g 12 ¼ g 21 ¼ 24
ð5:48Þ
and
3 Dg x ; xuu ¼ ðv þ sin uÞxv ; xuv ¼ 24 2 3 3 Dg x Dg x ¼ xv ; 24 ¼ 0: 24 2 u 2 v
xvv ¼ 0;
ð5:49Þ ð5:50Þ
Acknowledgments The first author would express his gratitude to professor Lihe Wang for his hospitality and helpful discussions during the first author’s visits to Shanghai Jiao Tong University and University of Iowa. The authors would like to thank the referees for their comments which have improved the paper presentation. This work was supported by the Fundamental Research Funds for the Central Universities (No. N130405006) and NSFC (Nos. 11201056 and 11371080), and the first author was financial supported by the China Scholarship Council. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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