# Tagged Questions

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such ...

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### Is this a normal form for $4$-forms on manifolds?

Starting from a $2$-form $\omega$ which is nondegenerate and closed on a $2n$-dimensional manifold, it is always possible to find local coordinates $x_1,y_1,\ldots,x_n,y_n$ so that the form $\omega$ ...
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### Degree of smooth maps are equal $\Rightarrow$ homotopic

It is an easily proven theorem that if $f,g:M\to N$ are smooth maps that are homotopic maps between compact, connected, oriented, smooth manifolds of dimension $n$, then $\deg f=\deg g$. I was ...
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### Differential forms in Physics

In the context of physics, I just read about the the symplectic 2-form $\omega$ on a symplectic manifold $M$ of dimension $2n$. Unfortunately, I could not follow a few arguments. I.e. it was said ...
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### I don't get the relationship between differentials, differential forms, and exterior derivatives.

I don't get the relationship between differentials, differential forms, and exterior derivatives. (Too many $d$'s getting me down!) Here are the relevant (partial) definitions from Wikipedia; ...