By Werner Hildbert Greub, Stephen Halperin, Ray Vanstone
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Additional resources for Connections, Curvature, and Cohomology: 2
A basis for the topology of X is a family 0 of open sets such that each open subset of X is the union of elements of 0. If 0 is closed under finite intersections, it is called an i-basis. If X has a countable basis, it is called second countable. s4. 12. Manifolds and vector bundles. , infinitely differentiable), second countable, Hausdorff, and finite dimensional. T h e set of smooth maps between manifolds M and N is written Y ( M ;N ) . + N has a smooth inverse, it is called a dzgeomorphism. Y ( M ) denotes the algebra of smooth real-valued functions on M .
An orientation of M is an orientation of T~ ; thus it is an equivalence class of nowhere vanishing n-forms. A smooth map v: M --t N (dim M = dim N ) is called orientation preserving (respectively, orientation reversing) if v*d (respectively, -v*d) represents the orientation of M when d represents that of N . , 4. Summary of volume I 19 in A ( M ) . Assume M oriented and of dimension n. Then the integral is defined; it is a linear map J M : A t ( M ) -+ R, natural with respect to orientation preserving diffeomorphisms, and satisfying where A , is the positive normed determinant function of an oriented Euclidean space E.
Conversely, assume cp is injective. Let V be a neighbourhood of 0 in Te(G) such that the restriction of exp, to V is injective. Then since exp, o cp' = 0 exp, , the restriction of expH 0 cp' to V is injective. I n particular, the restriction of p' to V is injective. Since cp' is linear and V is an open subset of Te(G),it follows that cp' is injective. Since each (dcp), is injective. Hence so is dcp. D. Corollary 111: If cp is bijective, then it is a diffeomorphism and hence an isomorphism between Lie groups.