Differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

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How to build smooth variations along arbitrary paths at the endpoint?

Let $M$ be a smooth manifold, $\gamma:[0,a] \to M$ a smooth path. Assume we are given another smooth path $\phi:(-\epsilon,\epsilon) \to M$ that starts from an endpoint of $\gamma$, i.e $\phi(0)=\...