Answer to Question #303506 in Differential Geometry | Topology for Jitendra

Question

1-(Invariance of dimension) Let M and N be smooth manifolds and suppose M is diffeo-

morphic to N. Then show that dim M = dim N.

2-(Inverse function theorem) Let M and N be smooth manifolds and let F : M → N be

smooth. Suppose DFp : TpM → TF(p)N is an isomorphism for each p . Then show that

M is locally diffeomorphic to N.

3-Let M and N be smooth manifolds and let πM : M × N → M and πN : M × N → N

be the projection maps. For any (p, q) ∈ M × N show that the map

Π : Tp(M × N) → TpM × TpN,

defined by

Π(v) = (D(πM)p(v), D(πN )q(v))

is an isomorphism.