Geodesics and Jacobi Fields
We start with a preliminary technical remark:
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Springer-Verlag Berlin Heidelberg GmbH
Jürgen Jost
Riemannian Geometry and Geometrie Analysis Second Edition
Springer
Jiirgen Jost Max Planck Institute for Mathematics in the Sciences Inselstr. 22-26 D-04103 Leipzig Germany
Mathematics Subject Classification (1991): 53B:n, 53C:w, 32.C17, 35!6, 49-XX, 58E2.o, 57R15
Library of Congress Cataloging-in Publication Data Jest, Jurgen, 1956Riemannlan geometry and gea•etrlc analysis 1 Jurgen Jost.2nd ed. p. em. -- fh n n2 Eo
(ii)
For any index set A : (ila)aEA C 0 =>
(iii)
U
aEA
ila
E0
0, ME 0
The sets from 0 are called open. A topological space is called Hausdorff if for any two distinct points P1, P2 E M there exists open sets il1, il2 E 0 with P1 E il11P2 E il2, il1 n il2 = 0. A covering (ila)aeA (A an arbitrary index set) is called locally finite if each p E M is contained in only finitely many ila: M is called paracompact if any open covering possesses a locally finite refinement. This means that for any open covering (ila)aeA there exists a locally finite open covering (ilp)t3eB with V {3 E B 3a E A:{}~ C ila.
A map between topological spaces is called continuous if the preimage of any open set is again open. A bijective map which is continuous in both directions is called homeomorphism. Definition 1.1.1 A manifold M of dimension d is a connected paracompact Hausdorff space for which every point has a neighborhood U that is homeomorphic to an open subset n of JRd. Such a homeomorphism x:U-til
is called a (coordinate) chart. An atlas is a family {Ua,xa} of charts for which the Ua constitute an open covering of M. Two atlantes are called compatible if their union is again an atlas. In general, a chart is called compatible with an atlas if adding the chart to the
2
1. Foundational Material
atlas yields again an atlas. An atlas is called maximal if any chart compatible with it is already contained in it.
Remarks. I) A point p E Ua is determined by xa(p); hence it is often identified with xa(p). Often, also the index a is omitted, and the components of x(p) E JRd are called local coordinates of p. 2)
Any atlas is contained in a maximal one, namely the one consisting of all charts compatible with the original one.
Definition 1.1.2 An atlas {Ua, Xa} on a manifold is called differentiable if all chart transitions
Xf3
0
x~ 1
:
Xa(Ua n U{3)
-t
X[3(Ua n U{3)
are differentiable of class coo (in case Ua n Uf3 :j:. 0). A maximal differentiable atlas is called a differentiable structure, and a differentiable manifold of dimension d is a manifold of dimension d with a differentiable structure.
Remarks. I)
Since the inverse of Xf3 o x~ 1 is Xa o xfj 1 , chart transitions are differentiable in both directions, i.e. diffeomorphisms.
2)
One could also require a weaker differentiability property than
3)
It is easy to show that the dimension of a differentiable manifold is uniquely determined. For a general, not differentiable manifold, this is much harder.
4)
Since any·differentiable atlas is contained in a maximal differentiable one, it suffices
