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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Need help understanding local coordinates of differential forms

I was trying to check a variation of theorem 2.2: I did the first part of the proof using the standard contact structure on $\mathbb R^{2n + 1}$ which worked out as expected. Then I wanted to do it ...
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An intuitive question about the metric $d(\cdot,\cdot)$ on a complete Riemannian manifold

Let $(M,g)$ be a complete Riemannian manifold. Suppose $\gamma_1,\gamma_2:[0,1] \to M$ are two different segments(i.e.minimizing geodesics) from $p$ to $x=\exp_p(v)$. Suppose further that $\gamma'_1(1)...
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What is the torsion and curvature in modern language?

I think it can not be the torsion and curvature in the connection context, for these two are anti-symmetric in two variables, so that they must vanish on a one-dimensional space (tangent space of a ...
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Is anyone talking about “ball bundles” of metric spaces?

In differential geometry: Each smooth manifold $M$ is equipped with a tangent bundle $TM,$ which is a manifold equipped with a projection back to $M$ Given a smooth map $f : M \rightarrow N$ between ...