Riemannian Manifolds
In this chapter we assume that the reader is familiar with the general concepts of Riemannian geometry. They may be found in many different books: [Cat], [Eis], [Ras], [BiC], [KoN], [Sp], [Wo], [GKM], [dCar], [GLP], [K 1]. Our notations are closer to [GKM
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In this ehapter we assume that the reader is familiar with the general eoneepts of Riemannian geometry. They may be found in many different books: [Cat] , [Eis], [Ras], [BiC], [KoN], [Sp], [Wo], [GKM], [dCar], [GLP], [K 1]. Our notations are closer to [GKM]. The variational theory of geodesies is used in an essential way. Its exposition may be found in the books [Mil 2], [Pos]. Comparison theorems are developed in part in [GKM], [K 1], [ChE], [BiC], [KoN]. For the sake of eonvenienee we will begin by reealling some definitions.
§ 31. Principal Notions 31.1. Notations. Suppose M is a Riemannian manifold of dimension m ~ 2, while N is a smooth submanifold of dimension n; m > n ~ 0; let T"M, T"N be the tangent spaees at the point x; the braekets ( , ) denote the sealar produet in T"M; i: N ~M is the inclusion map; di: T,.,N -+ T"M is the differential of i. We shall always assurne that di(T,.,N) is eanonieally identified with T,.,N. If Yis a given veetor field on M in a neighbourhood of the point XE M, while XE T,.,M, then Vx Y denotes, as usual, the eovariant derivative. The orthogonal projeetion of the veetor XE T"M into T"N, XE N, is denoted by X T • 31.2. The Second Fundamental Form. 31.2.1. Suppose dirn N = n > 0 and let v,,(N) be the orthogonal eomplement to T"N in T"M. The second fundamental form of the submanifold N at the point x is the map h: T"N x T"N -+ v,,(N) defined by the relation h(X, Y) = Vx Y - (VX Y)T.
(1)
This is weIl defined, since h(X, Y), as ean be shown, depends only on the value Y(x) and not on the ehoice of the field Y. For Z E vAN) we put hz(X, Y) = (h(X, Y),Z) = (VxY,Z),
(2)
hand hz being symmetrie bilinear forms; his a form with values in v,,(N), hz is a sealar form. Using the bilinear symmetrie form hz , we ean eonstruet in a unique way the Y. D. Burago et al., Geometric Inequalities © Springer-Verlag Berlin Heidelberg 1988
233
§31. Principal Notions
linear transformation hz : T,.N -+ TxN such that
0, then
V(M) ::(
W m- n- 1 V(N)
I
R 0
(sinhj"=kt)m-n-l j"=k (coshj"=ktt dt
(8)
34.1.8. Corollary. Under the assumptions of Theorem 34.1.2, if N is a closed geodesie of length L (in this case n = 1, h = 0), then, taking into consideration the relation {m - l)w m = 2nw m - 2 , we can rewrite (1) in the form
250
Chapter 6. Riemannian Manifolds
where R* = R, when k ~ 0 and R* = min{n/2Jk,R}, when k > O. Suppose ri is the injectivity radius of the c10sed manifold M. It is known [GKMJ that Ku ~ K implies either K > 0 and ri ~ n/Jk, or there exists a non-trivial periodic geodesic of minimal length L = 2ri • Hence (9) implies the following 34.1.9. CoroUary. Suppose that under the assumptions of Theorem 34.1.2, k ~ Ku ~ K. Then
min
{fi, :m
min {~
(Jk)m-l V(M)}
~ ( Fk
.)K'wm
n ( --
sinhFkR
)m-l V(M)}
Fk )m-l V(M)
w m sinhFkR
when k > 0, whenk
~O
whenK
~
0, when k = 0,
n
(sinhyCkr)l-n
J:
(sinhyCkt)n-l dt
(12)
when k < 0;
It is assumed that n ~ 2, r > 0 and, in the case k > 0, that Jkr < n. This function, expressing the ratio of the volume of the n-dimensional b
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