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How to prove the following property? I cannot do anything.

Let $M$ be a connected paracompact smooth manifold of dimension $m\geq 2$. Let $(p_k), (q_k)_{k\in \mathbb{N}}$ be sequences on $M$ which do not accumulate on anywhere and $i\neq j \Rightarrow p_i\neq p_j, q_i\neq q_j$. Let $X_k\in (T_{p_k}M)\backslash\{0\}, Y_k\in (T_{q_k}M)\backslash\{0\}$. Then there is a diffeomorphism $\varphi :M\to M$ with \begin{equation} \varphi(p_k)=q_k,\ \ \ d\varphi_{p_k}(X_k)=Y_k\ \ \ (k\in\mathbb{N}). \end{equation}

stb2084
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  • The case of pairs of points (without the condition on the differential) is the topological homogeneity of connected manifolds, for example see here http://math.stackexchange.com/questions/89721/are-all-connected-manifolds-homogeneous . Once you've proven that you're almost done... – Emilio Ferrucci Aug 03 '14 at 18:05

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