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§31 Normals from a Point to a Manifold
§ 3' NORMALS FROM A POINT TO A MANIFOLD
Let M , be a regular compact C"-manifold in Em with 0 < n < m. In Theorem 15.3 we have seen that the focal points of M , in Em are nowhere dense in E m . Let q be a point fixed in Em - M, and p an arbitrary point in M , . We term the mapping
the distance function f , with pole q and domain M, . By definition of a focal point the function f , is ND if and only if q is not a focal point of M , . Let (q, 5) be a straight arc orthogonal to M , at a point 5 EM, . We term (q, 5) an arc normal to M , at 5 and assign this arc an index equal to the index of 5 as a critical point off, . Theorem 30.1 then yields the following:
Em - M , not a focalpoint of M , , and for i on the range 0, 1, ...,n let mi be the number of normal arcs (q, 5) of index i, counting arcs (q, 5) as dazerent if their base points 5 are dz#erent. If R, is the ith connectivity of M , , the relations (30.1) are satisfed, as are the relations Theorem 31.1. Let q be apoint of
mi>Ri,
i=O
,...,n.
(31.2)
An Example. Let M,be a torus in E 3 . The focal points of M , are
r
the points on the axis of the torus and on the central circle of the corresponding solid torus. For the torus R, = R, = 1 and R, = 2, 281
282
Iv. OTHER
APPLICATIONS OF CRITICAL POINT THEORY
as we shall see. If q is not a focal point of the torus, nor on the torus, there are four straight arcs (q, [), normal to M , at points 5 of M , . If q is in the plane of and exterior to the circle of M , of maximum length, each of these normal arcs is a subarc of the longest of these arcs. Given a normal arc (q, t),we shall evaluate the index of 5 as a critical point of the distance function f, , beginning with the case in which m=n+l. In making this evaluation we shall admit translations or orthogonal transformations of the rectangular coordinates of Em.It is understood that a point p E M , and the pole q in (31.1) undergo the same changes of coordinates, so that if x = (x, ,...,x,) and a = (a, ,..., a,) represent, respectively, p and q, then 11 x - a 11 is invariant under any admissible change of coordinates.
r
.
+
The Case m = n + 1. Centers of Curvature of M,, When m = n 1 we shall evaluate the index of a critical point 5 E M , of the distance function f, in terms of the centers of principal normal curvature of M, on the normal to M , at 5. Such centers must be defined. Given a straight arc (q, 5) normal to M , at 5, let a system of rectangular coordinates x be chosen in Em such that 5 E M , is represented by the origin, and the pole q is represented by the point (0,..., 0, c) E Em with (n 1)st coordinate c = 11 q - 5 11. Setting (x, ,...,x,) = (0, ,...,v,), there will exist for 11 w 11 sufficiently small a Monge presentation (31.3) %+l= (%/2) v j
+
++
of a neighborhood of the point 5 on M , ,in which ai5wtvjis a symmetric throughout this quadratic form and the remainder (indicated by proof) is a function w +L(v) of class C", vanishing with its first and second partial derivatives at the origin. Let p, ,pa ,...,pn be the characteristic roots of the matrix 11 aU 11. There then exists an orthogonal transformation of the coordinates w, ,..., w, into coordinates u, ,..., u, ,by virtue of which a neighborhood of 5 on M , has a Monge presentation
++
= Q(p&
%I+,
for 11 u 11
+ + '.*
< e and e sufficiently small.
Pn'Ln2)
++
(31.4)
31. NORMALS FROM
A POINT TO A MANIFOLD
283
If the coordinates u, are properly numbered and the roots pi correspondingly numbered,.then for some 7 on the range 0, 1,..., n (31.5)
Corresponding to each characteristic root Ph # 0 we set 8 h = 1/ph and term &?h a radius of principal normal curvature belonging to the point [ on M,. The point Ph on the normal A, to M , at [ whose coordinate xn+, = &?h will be called a center of principal normal curvature of M , belonging to [. Such a center will be counted with a multiplicity equal to the multiplicity of Ph as a characteristic root of 11 aijII. One sees that the centers P, ,...,P, of principal normal curvature on the normal A, to M , at [ are uniquely determined, except for order, by M , and [ E M, . We shall prove the following theorem:
+
Theorem 31.2. Suppose that m = n 1 and that (9, [) is a straight arcjoining q E Em - M , to [ E M , , orthogonal to M , at [, with [ a N D critical point of the distance function f , . The index of [ as a critical point o f f , is then the number of centers of principal normal curvature of M , belonging to [ on the open arc (4, [), counting these centers with their multiplicities.
T o determine the index of [ as a critical point of the distance function f , , we make use of the representation (31.4) of M , near [, denoting the right side of (31.4) by cp(u). In terms of the coordinates (x, ,...,x,+,) = (ul,..., u,, xn+,) employed in (31.4) the pole q has a = (0,..., 0, c), with c > 0. Hence for representation (a, ,...,a,,) points x E M , with coordinates (x1 3 . . . , %+l) = (ill Y...,
11 x - a 112 - 8
= 11 #
= (1 - p1c)u12
112
+ 1 cp(u) - c
12
un
- c2
+ + (1 - p T c ) ~ ; +
(31.6)
* cp(u)),
+ + ++. U;
(31.7)
The critical point [ off, is also a critical point of fq2on M , . Since f,([) # 0, one sees that f , and fqahave the same index at their common critical point
[.
284
Iv.
OTHER APPLICATIONS OF CRITICAL POINT THEORY
The index of the quadratic form (31.7) is thus the index of 5 as a critical point off, This index is clearly the number of the characteristic roots p h # 0 for which 1 - phc < 0, or, equivalently, the number of the positive radii w h < c. Finally, no @h = c, since, by hypothesis q is not a focal point of M , , so that 5 is a ND critical point off, , or, equivalently, (31.7) is a ND quadratic form. Hence 1 - phc # 0 for h = 1, 2,..., I , implying that no = c. Theorem 31.2 follows.
.
Remark. Suppose, contrary to the hypothesis of Theorem 31.2, that the pole q is a center of principal normal curvature based on the point 5 E M , . Then, for some h on the range 1,...,I , 1 - phc = 0 and the multiplicity of the center q, as defined above, is the multiplicity of p h as a characteristic root. One sees that this multiplicity of ph is the nullity of the form (31.7), or, equivalently, the nullity p of the critical point 5. The Case m > n. An Extension of Theorem 31.2. A point q E Em- M , has been called a focal point of M , based on a point 5 E M , if 5 is a degenerate critical point of the distance function f,
.
Definition 31.1. The nullity p of a focal point q of M , based on a point 1 of M, is by definition the nullity of 5 as a criticalpoint off, . The following theorem extends Theorem 31.2. It concerns the general case m > n as distinguished from the special case m = n 1.
+
Suppose that m > n and that (4, 5) is a straight arc to 5 E M , , with 5 a ND critical point off, The index of 5 as a m'tical point of f, is then the number of focal points of M , belonging to 5 on the open arc (q, 5), counting these focal points with their nullities. Theorem 31.3.
.
porn q E Em - M ,
+
Proof when m = n 1. In this case Theorem 31.3 is no more than a reinterpretation of Theorem 31.2 taking account of Definition 31.1 of the nullity of a focal point q of M , In fact, the Remark following the proof of Theorem 31.2 has the following consequence:
.
31.
NORMALS FROM A POINT TO A MANIFOLD
285
+
When m = n 1 a center q E En+, - M , of principal normal curvature of M . is “based” on a degenerate critical point l, in M , and so is a focal point of M , based on 5. The multiplicity of q as a center of principal normal curvature based on 1, is by Definition 31.1 the nullity of q as a focal point of M , based on l,. Thus when m = n 1 Theorems 31.2 and 31.3 are equivalent.
+
Method of Proof of Theorem 31.3 in the General Case. Theorem 31.3 belongs to a class of theorems in global variational theory (Morse [131) capable of a proof by special methods involving “broken primary and secondary extremals.” It will be proved in a sequel to the present book concerned with this variational theory. The proof involves an examination of the variation of the “index” of a “critical extremal” [here the arc (4, 1,)] as the endpoints of the extremal vary.
A Second Proof of Theorem 31.3 Consider the case in which m>n+l. Notation. Suppose that the critical point 1, is at the origin of coordinates in Em and that the n-plane En of coordinates x1 ,..., x, is tangent to M , at l,. Suppose further that the pole q is on the x,+~axis with xn+l = c > 0. For 11 w 11 sufficiently small a neighborhood N of 1, relative to M , admits a Monge presentation (XI ,.s
xn) = (wl ,. *,
x,
=(
wn)
~ 3 2wiwj )
++,
a =
I ,...,m - n.
(31.8)
Let the coordinates (xl ,...,x,) = (wl ,..., en) be orthogonally transformed into coordinates (ul ,..., un) such that (31.4) holds as before. We adopt the notation of (31.5). In terms of the new rectangular coordinates u1 ,..., u, the neighborhood N of the origin, relative to M , , admits a Monge presentation replacing the Monge presentation (31.8). With only a slight additional complexity of reasoning one finds that at a point p(u) E N, represented by u,f,2(p(u)) - c2 is equal to the form (31.7) plus a remainder R(u) of the same character as in the case m = n 1. The index of 1,, as a critical point off, or f2, is the number of roots Ph # 0 for which 1 - cph < 0, counting these roots with their multiplicities as characteristic roots of the matrix 11 a t 11.
+
286
Iv.
OTHER APPLICATIONS OF CRITICAL POINT THEORY
An Interpretation. Let c’ be a positive number and let q’ be the point on the x,+~ axis at which x,+~ = c’. If 1 - phc’ = 0 for some h on the range 1,...,r , the form (31.7), with c’ replaced by c, is degenerate. That is, by Definition 15.1 the point q’ is a focal point of M,, belonging to 5. If p,, has the multiplicity p as a characteristic root of 1) u& 11, we see that the form (31.7), with c replaced by c’, has the nullity p, since just p of the tt terms in (31.7) would then vanish. By virtue of Definition 31.1 the focal point q’ is “counted” with a nullity p. The index of the form 31.7 is thus the number of focal points of M , “belonging to 5” on the open arc (q, t), counting these focal points with their nullities. The interpretation of focal points in the variational theory is nearer their interpretation in geometric optics.
Let M , be a regular compact C"-manifold in Em with 0 < n < m. In Theorem 15.3 we have seen that the focal points of M , in Em are nowhere dense in E m . Let q be a point fixed in Em - M, and p an arbitrary point in M , . We term the mapping
the distance function f , with pole q and domain M, . By definition of a focal point the function f , is ND if and only if q is not a focal point of M , . Let (q, 5) be a straight arc orthogonal to M , at a point 5 EM, . We term (q, 5) an arc normal to M , at 5 and assign this arc an index equal to the index of 5 as a critical point off, . Theorem 30.1 then yields the following:
Em - M , not a focalpoint of M , , and for i on the range 0, 1, ...,n let mi be the number of normal arcs (q, 5) of index i, counting arcs (q, 5) as dazerent if their base points 5 are dz#erent. If R, is the ith connectivity of M , , the relations (30.1) are satisfed, as are the relations Theorem 31.1. Let q be apoint of
mi>Ri,
i=O
,...,n.
(31.2)
An Example. Let M,be a torus in E 3 . The focal points of M , are
r
the points on the axis of the torus and on the central circle of the corresponding solid torus. For the torus R, = R, = 1 and R, = 2, 281
282
Iv. OTHER
APPLICATIONS OF CRITICAL POINT THEORY
as we shall see. If q is not a focal point of the torus, nor on the torus, there are four straight arcs (q, [), normal to M , at points 5 of M , . If q is in the plane of and exterior to the circle of M , of maximum length, each of these normal arcs is a subarc of the longest of these arcs. Given a normal arc (q, t),we shall evaluate the index of 5 as a critical point of the distance function f, , beginning with the case in which m=n+l. In making this evaluation we shall admit translations or orthogonal transformations of the rectangular coordinates of Em.It is understood that a point p E M , and the pole q in (31.1) undergo the same changes of coordinates, so that if x = (x, ,...,x,) and a = (a, ,..., a,) represent, respectively, p and q, then 11 x - a 11 is invariant under any admissible change of coordinates.
r
.
+
The Case m = n + 1. Centers of Curvature of M,, When m = n 1 we shall evaluate the index of a critical point 5 E M , of the distance function f, in terms of the centers of principal normal curvature of M, on the normal to M , at 5. Such centers must be defined. Given a straight arc (q, 5) normal to M , at 5, let a system of rectangular coordinates x be chosen in Em such that 5 E M , is represented by the origin, and the pole q is represented by the point (0,..., 0, c) E Em with (n 1)st coordinate c = 11 q - 5 11. Setting (x, ,...,x,) = (0, ,...,v,), there will exist for 11 w 11 sufficiently small a Monge presentation (31.3) %+l= (%/2) v j
+
++
of a neighborhood of the point 5 on M , ,in which ai5wtvjis a symmetric throughout this quadratic form and the remainder (indicated by proof) is a function w +L(v) of class C", vanishing with its first and second partial derivatives at the origin. Let p, ,pa ,...,pn be the characteristic roots of the matrix 11 aU 11. There then exists an orthogonal transformation of the coordinates w, ,..., w, into coordinates u, ,..., u, ,by virtue of which a neighborhood of 5 on M , has a Monge presentation
++
= Q(p&
%I+,
for 11 u 11
+ + '.*
< e and e sufficiently small.
Pn'Ln2)
++
(31.4)
31. NORMALS FROM
A POINT TO A MANIFOLD
283
If the coordinates u, are properly numbered and the roots pi correspondingly numbered,.then for some 7 on the range 0, 1,..., n (31.5)
Corresponding to each characteristic root Ph # 0 we set 8 h = 1/ph and term &?h a radius of principal normal curvature belonging to the point [ on M,. The point Ph on the normal A, to M , at [ whose coordinate xn+, = &?h will be called a center of principal normal curvature of M , belonging to [. Such a center will be counted with a multiplicity equal to the multiplicity of Ph as a characteristic root of 11 aijII. One sees that the centers P, ,...,P, of principal normal curvature on the normal A, to M , at [ are uniquely determined, except for order, by M , and [ E M, . We shall prove the following theorem:
+
Theorem 31.2. Suppose that m = n 1 and that (9, [) is a straight arcjoining q E Em - M , to [ E M , , orthogonal to M , at [, with [ a N D critical point of the distance function f , . The index of [ as a critical point o f f , is then the number of centers of principal normal curvature of M , belonging to [ on the open arc (4, [), counting these centers with their multiplicities.
T o determine the index of [ as a critical point of the distance function f , , we make use of the representation (31.4) of M , near [, denoting the right side of (31.4) by cp(u). In terms of the coordinates (x, ,...,x,+,) = (ul,..., u,, xn+,) employed in (31.4) the pole q has a = (0,..., 0, c), with c > 0. Hence for representation (a, ,...,a,,) points x E M , with coordinates (x1 3 . . . , %+l) = (ill Y...,
11 x - a 112 - 8
= 11 #
= (1 - p1c)u12
112
+ 1 cp(u) - c
12
un
- c2
+ + (1 - p T c ) ~ ; +
(31.6)
* cp(u)),
+ + ++. U;
(31.7)
The critical point [ off, is also a critical point of fq2on M , . Since f,([) # 0, one sees that f , and fqahave the same index at their common critical point
[.
284
Iv.
OTHER APPLICATIONS OF CRITICAL POINT THEORY
The index of the quadratic form (31.7) is thus the index of 5 as a critical point off, This index is clearly the number of the characteristic roots p h # 0 for which 1 - phc < 0, or, equivalently, the number of the positive radii w h < c. Finally, no @h = c, since, by hypothesis q is not a focal point of M , , so that 5 is a ND critical point off, , or, equivalently, (31.7) is a ND quadratic form. Hence 1 - phc # 0 for h = 1, 2,..., I , implying that no = c. Theorem 31.2 follows.
.
Remark. Suppose, contrary to the hypothesis of Theorem 31.2, that the pole q is a center of principal normal curvature based on the point 5 E M , . Then, for some h on the range 1,...,I , 1 - phc = 0 and the multiplicity of the center q, as defined above, is the multiplicity of p h as a characteristic root. One sees that this multiplicity of ph is the nullity of the form (31.7), or, equivalently, the nullity p of the critical point 5. The Case m > n. An Extension of Theorem 31.2. A point q E Em- M , has been called a focal point of M , based on a point 5 E M , if 5 is a degenerate critical point of the distance function f,
.
Definition 31.1. The nullity p of a focal point q of M , based on a point 1 of M, is by definition the nullity of 5 as a criticalpoint off, . The following theorem extends Theorem 31.2. It concerns the general case m > n as distinguished from the special case m = n 1.
+
Suppose that m > n and that (4, 5) is a straight arc to 5 E M , , with 5 a ND critical point off, The index of 5 as a m'tical point of f, is then the number of focal points of M , belonging to 5 on the open arc (q, 5), counting these focal points with their nullities. Theorem 31.3.
.
porn q E Em - M ,
+
Proof when m = n 1. In this case Theorem 31.3 is no more than a reinterpretation of Theorem 31.2 taking account of Definition 31.1 of the nullity of a focal point q of M , In fact, the Remark following the proof of Theorem 31.2 has the following consequence:
.
31.
NORMALS FROM A POINT TO A MANIFOLD
285
+
When m = n 1 a center q E En+, - M , of principal normal curvature of M . is “based” on a degenerate critical point l, in M , and so is a focal point of M , based on 5. The multiplicity of q as a center of principal normal curvature based on 1, is by Definition 31.1 the nullity of q as a focal point of M , based on l,. Thus when m = n 1 Theorems 31.2 and 31.3 are equivalent.
+
Method of Proof of Theorem 31.3 in the General Case. Theorem 31.3 belongs to a class of theorems in global variational theory (Morse [131) capable of a proof by special methods involving “broken primary and secondary extremals.” It will be proved in a sequel to the present book concerned with this variational theory. The proof involves an examination of the variation of the “index” of a “critical extremal” [here the arc (4, 1,)] as the endpoints of the extremal vary.
A Second Proof of Theorem 31.3 Consider the case in which m>n+l. Notation. Suppose that the critical point 1, is at the origin of coordinates in Em and that the n-plane En of coordinates x1 ,..., x, is tangent to M , at l,. Suppose further that the pole q is on the x,+~axis with xn+l = c > 0. For 11 w 11 sufficiently small a neighborhood N of 1, relative to M , admits a Monge presentation (XI ,.s
xn) = (wl ,. *,
x,
=(
wn)
~ 3 2wiwj )
++,
a =
I ,...,m - n.
(31.8)
Let the coordinates (xl ,...,x,) = (wl ,..., en) be orthogonally transformed into coordinates (ul ,..., un) such that (31.4) holds as before. We adopt the notation of (31.5). In terms of the new rectangular coordinates u1 ,..., u, the neighborhood N of the origin, relative to M , , admits a Monge presentation replacing the Monge presentation (31.8). With only a slight additional complexity of reasoning one finds that at a point p(u) E N, represented by u,f,2(p(u)) - c2 is equal to the form (31.7) plus a remainder R(u) of the same character as in the case m = n 1. The index of 1,, as a critical point off, or f2, is the number of roots Ph # 0 for which 1 - cph < 0, counting these roots with their multiplicities as characteristic roots of the matrix 11 a t 11.
+
286
Iv.
OTHER APPLICATIONS OF CRITICAL POINT THEORY
An Interpretation. Let c’ be a positive number and let q’ be the point on the x,+~ axis at which x,+~ = c’. If 1 - phc’ = 0 for some h on the range 1,...,r , the form (31.7), with c’ replaced by c, is degenerate. That is, by Definition 15.1 the point q’ is a focal point of M,, belonging to 5. If p,, has the multiplicity p as a characteristic root of 1) u& 11, we see that the form (31.7), with c replaced by c’, has the nullity p, since just p of the tt terms in (31.7) would then vanish. By virtue of Definition 31.1 the focal point q’ is “counted” with a nullity p. The index of the form 31.7 is thus the number of focal points of M , “belonging to 5” on the open arc (q, t), counting these focal points with their nullities. The interpretation of focal points in the variational theory is nearer their interpretation in geometric optics.