Riemannian Geometry
A manifold M n , of dimension n, is a Haussdorff topological space such that each point of M n , has a neighborhood homeomorphic to ℝ n . Thus a manifold is locally compact and locally connected. Hence a connected manifold is pathwise connected.
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Riemannian Geometry
§1. Introduction to Differential Geometry 1.1 A manifold M", of dimension n, is a Haussdorff topological space such that each point of M" has a neighborhood homeomorphic to IR". Thus a manifold is locally compact and locally connected. Hence a connected manifold is pathwise connected. 1.2 A local chart on M" is a pair (0, cp), where a is an open set of M" and cP a
homeomorphism of a onto an open set of IR". A collection (ai' CPi)i e I of local charts such that Ui e I OJ = M" is called an atlas. The coordinates of Pea, related to cP, are the coordinates of the point cp(P) of IR".
1.3 An atlas of class Ck (respectively, c ~, CW ) on Mil is an atlas for which all changes of coordinates are C k (respectively, CX, C W ). That is to say, if (O:Z' CP2) and (all' CPII) are two local charts with O2 11 Op :1= 0, then the map CP7. s cpi 1 of CPII(fJ.:z 11 all) onto cp:z(O:z 11 Oil) is a diffeomorphism of class Ck (respectively, Coo, CW). 1.4 Two atlases of class Ck on Mil (Uj, CPi)iel and (~, "':Z)lI.eA are said to be
equivalent if their union is an atlas of class Ck. By definition, a differentiable manifold of class Ck (respectively, Coo, CW) is a manifold together with an equivalence class of Ck atlases, (respectively, Coo, CW). 1.5 A mapping f of a differentiable manifold Ck : Wp into another M", is called differentiable C (r ::;; k) at P e U c Wp if'" fo qJ -1 is differentiable C at cp(P), and we define the rank ofJ at P to be the rank of", J cp - 1 at qJ(P). Here (U, cp) is a local chart of Wp and (0. "') a local chart of Mil with f(P)eO. 0
0
0
A C differentiable mappingf is an immersion if the rank ofJ is equal to p for every point P of Wp. It is an imbedding ifJis an injective immersion such that J is a homeomorphism of Wp onto JCWp) with the topology induced from Mil.
T. Aubin, Some Nonlinear Problems in Riemannian Geometry © Springer-Verlag Berlin Heidelberg 1998
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I. Riemannian Geometry
1.1. Tangent Space 1.6 Let (0, qJ) be a local chart and f a differentiable real-valued function defined on a neighborhood of P E O. We say that f is flat at P if d(f 0 qJ - 1) is zero at qJ(P). A tangent vector at P EM" is a map X:f -+ X(f) E IR defined on the set of functions differentiable in a neighborhood of P, where X satisfies: (a) (b) (c)
If A, j.l E IR, X(Af + j.lg) = AX(f) + j.lX(g). X(f) = 0, iffis flat. X(fg) = f(P)X(g) + g(P)X(f); this follows from (a) and (b).
1.7 The tangent space Tp(M) at P E Mn is the set of tangent vectors at P. It has a natural vector space structure. In a coordinate system {Xi} at P, the vectors (O;OXi)p defined by (O;OXi)p (f) = [c(fo qJ-l)/OXi] 0 if X =I
o.
Hereafter, unless otherwise stated, a Riemannian manifold Mil is a connected Coo Riemannian manifold of dimension n. 1.16 Theorem. On a paracompact Coo differentiable manifold, there exists a Coo Riemannian metric g.
Proof Let (OJ, lfJJiel be an atlas and {tXJ a COO partition of unity subordinate to the covering {OJ Such {tXJ exists since the manifold Mil is paracompact. Set 8 = (8 ik ) be the Euclidean met
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