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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closed form is exact in euclidean space

Question is to show that $d(f)=0$ for a $0$ form on $\mathbb{R}^n$ then $f$ is a constant function. See that $$0=df=\sum_i\frac{\partial f}{\partial x_i}dx_i$$ implies that $\frac{\partial ...
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Index of zero of a gradient vector field at a critical point

Let $M$ be a Riemannian manifold with a Morse function $f: M \to \mathbb{R}$. The zeroes of the gradient vector field of $f$ are the critical points of $f$. How do you show that a critical point ...