By Richard L Bishop

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F) Z mn iFm = 0. (g) y m = *m + 2,. 4. , d m = {h I h €Zm}. This defines a one-to-one real linear map P : d m +Z m , t --t h. + + Let j : Y m -+ Y mbe multiplication by i. Define J = P-l jP. (i) J z = -identity. (j) Compute J in terms of real coordinate vector fields which come from the real and imaginary parts of a complex coordinate system. (k) J is defined on T ( M ) and is a bundle map. An almost complex structure on a manifold M is a bundle map J : T ( M ) + T ( M ) such that (1) J ( M m ) = M , for every m E M .

5 Exponential Map Let G be a Lie group, X E ~Let . y x be the integral curve of X starting at the identity. 5. Exponential Map which assigns yx(l) to X ; we write exp t + exp t X is just y x . X = yx(l). Clearly the map FIG. 13. Commutativity with homomorphisms. then the diagram I7 If j : G +H is a homomorphism, dj exp exp is commutative. G-H Proof. This follows immediately from the fact that y d j ( x ) = j o y x for any X E ~ . From this and the following theorem we see that the exponential map gives the correspondence between subalgebras of g and subgroups of G.

1121. However, if G is simply connected, a local homomorphism can be extended to a homomorphism, so we obtain: Let G and H be Lie groups with G simply connected. Then the correspondence j t)d j is one-to-one between homomorphisms of G into H and homomorphisms of g into b. Theorem 2. T h e kernel of a homomorphism is a closed normal subgroup, and the kernel of the corresponding Lie algebra homomorphism is an ideal, and it is easily seen that this ideal belongs to the subgroup. Conversely, if H is a closed normal subgroup of G, then the set of left cosets G / H can be given a natural manifold structure in such a way that the projection G -+ G / H is a homomorphism of Lie groups.

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Geometry of Manifolds by Richard L Bishop


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