# 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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### Find surface in $\mathbb{R}^3$ with certain tangent spaces

By Frobenius Theorem, in $\mathbb{R}^3$ there exists a smooth surface whose tangent space is spanned by the vector fields $V(x,y,z)=(x^2+y^2,0,-y)$ and $W(x,y,z)=(0,x^2+y^2,x)$. How can I find this ...
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### Can a non-zero vector field have zero divergence and zero curl?

I don't see how. Curl and divergence are essentially "opposites" - essentially two "orthogonal" concepts. The entire field should be able to be broken into a curl component and a divergence component ...
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### Covariant Taylor series

I am reading the following lecture notes of Avramidi https://www.researchgate.net/publication/255565392_Analytic_and_geometric_methods_for_heat_kernel_applications_in_finance I want to understand ...
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### Misunderstanding of Atiyah-Singer

I just looked up the Atiyah-Singer theorem and by ignoring technical details I had the impression that it tells us that any elliptic operator on a compact manifold satisfies Analytical index = ...
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### Can terms in an integrand cancel terms in the volume element of the integral?

More specifically, given an integrand $\phi(x^{0},x^{1},x^{2},x^{3})$ of the form: $$\phi(x^{0},x^{1},x^{2},x^{3})=\frac{1}{\sqrt{\mid g_{00}\mid}}e_{0}\psi(x^{0},x^{1},x^{2},x^{3})$$ Where the ...
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### Norm of the gradient of a vector field in Cartesian versus Cylindrical coordinates

It is well known that for a vector $\textbf{v}=R\textbf{e}_r+\Theta\textbf{e}_{\theta}+Z\textbf{e}_z$, its 2-norm is $\|\textbf{v}\|_2=\sqrt{R^2+Z^2}$ instead of $\sqrt{R^2+\Theta^2+Z^2}$. Now, for a ...
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### Covariant derivative of parallel transport

I am learning Riemannian geometry and don't get why the following is true. We are on a Riemannian manifold with the Levi Cevita connection $\nabla$. Let $\mathcal{P}(x,x')$ be the parallel transport ...
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### Intuitive interpretation of Ricci Flow

What is the best way to interpret, explain or somehow visualize the basic idea behind formal definition of Ricci Flow? I am familiar with the hackneyed expressions like "Ricci Flow is a non-...
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### surface gradient of the unit normal vector

I am reading a book that defines a curvature tensor as $\boldsymbol{K} = -\nabla_\pi \boldsymbol{\hat{n}}$ where $\boldsymbol{\hat{n}}$ is the unit normal vector of a surface and \begin{align*} \...
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### A question on the non trivial rank three bundle on $S^2$

We know that number of rank $3$ vector bundles on $S^2$ is just the number of equivalence classes of maps from $S^1$ to $SO(3)$. This implies that there is one and only one non trivial vector bundle ...
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### Integration of differential form on ellipsoidal surface with singularity in origin

As picture below ,I want to compute the (2) , because there is a singularity in $\{0\}$ and $\omega$ is closed . So ,I have $$\int_M\omega=\int _{\partial B_1(0)} \omega$$ I think there is a ...
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### Invariant connection form

Let $\pi : E \to \mathbb{R}^3$ define a vector bundle with a connection form $\nabla$ on $\mathbb{R}^3$. The text that i'm am reading then goes on to say that $\nabla$ is $\mathbb{R}$-invariant in ...