# 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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### Klein Bottle embedding on $\mathbb{R}^4$ [closed]

Prove that no embedding of a Klein bottle in $\mathbb{R}^4$ can be given by a system of equations $f_1=0, f_2=0$ with independent smooth functions $f_1,f_2$ (i.e. where $df_1,df_2$ are linearly ...
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### Green's Function for Laplacian on $S^1 \times S^2$

As indicated by the title, I am looking to find the Green's function for the Laplacian on $S^1 \times S^2$. Is such a function known? If not, does anyone have an approach to constructing such a ...
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### Principal null Direction

I need to understand what the principal null dierctions are in mathematics. Physicists define a principal null direction in a spacetime as a null vector which satisfies the Penrose-Debever equation. I ...
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### Restriction of a $C^{\infty}$ vector bundle over a regular submanifold.

This question is about the content of page 134 of Tu's An Introduction to Manifolds. A $C^{\infty}$ vector bundle or rank r is a triple $(E,M,\pi)$ consisting of manifolds $E$ and $M$ and a ...
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### Symbol of differential operator and change of coordinates

Some time ago I posted the question about the change of coordinates in differential operator. Here is the relevant discussion Symbol of differential operator transforms like a cotangent vector The ...
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### Trajectories of a vector field on the 2-sphere

Consider the vector field given by given by $(-zx,zy,0)$, where we've identified $T_pS^2$ where we've identified the space of vectors orthogonal to $p$. How do we visualize the trajectories of the ...
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### Perturbing a regular submanifold to ensure submersion.

Suppose $M$ is a regular compact submanifold with boundary of dimension $n$ in $\mathbb R^{n+1}\smallsetminus 0$, and assume that for each $v\in S^n$ the ray emanating from $v$ intersects $M$ in at ...
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### Product of currents

De Rham currents are for differential forms as distributions are to (smooth) functions. There is a notion of exterior product for differential forms: I wonder whether there is an algebra structure on ...
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### Sections of tensor bundle are tensor product of sections

Given $E,F$ vector bundles over a manifold $M$, I would like to know a proof of $\Gamma(E\otimes F) = \Gamma(E) \otimes_{C^\infty(M)} \Gamma(F)$. Where $\Gamma$ denotes the smooth sections over $M$. ...
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### Vector bundles and de Rham cohomology

So $M$ is a compact manifold and I am asked to either prove the following statement or give a counterexample: if $\pi: E \rightarrow M$ is a vector bundle, then $H^2(E) \simeq H^2(M)$. I know the ...
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### Differentiable functions between manifolds are continuous

Let $f:M \to N$ be differentiable function between manifolds. I want to show that $f$ is continuous. First, that $f$ is continuous should mean (correct me if I'm wrong!) that for every point $a\in N$ ...
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### How to parallel transport a coframe field in a geodesic normal neighborhood?

From Chern: Lectures on Differential Geometry, page 147 Chern claimed that a torsion-free connection is completely determined locally by the curvature tensor. To show that he considered a geodesic ...
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### Is this the correct way to compute tensor bundles of smooth manifolds given by a smooth function?

Let $M$ be a smooth submanifold of $\mathbb{R}^n$ given by the vanishing locus of a smooth function $f(x_1,\ldots,x_n)$. I can compute the cotangent bundle from this embedding by looking at the ...
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### Construction of connections given in “From Calculus to Cohomology”

I'm struggling to understand part of the descripition of a connection on page 167 of Madsen, Tornehave "From Calculus to Cohomology". Immediately after defining a connection $\nabla$ on a bundle $\xi$ ...
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### Intuition about antisymmetrizing tensor equations

I was looking at the symmetries of the Riemann tensor, and tried to prove a couple of properties, namely If $\nabla$ is torsion-free, then: (i) $R^a_{\,[bcd]}=0$, and (ii) $R^a_{\,b[cd;e]}=0$. ...
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### Velocity of a 2-parameter curve

Let $M$ be a manifold and $I,J$ be two intervals on $\mathbb{R}$. Suppose $\alpha:I\times J\longrightarrow M$ a smooth map. It is clear that $s\mapsto\alpha(s,t_0)$ and $t\mapsto \alpha(s_0,t)$ are ...
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### Maurer Cartan Form of the Heisenberg group

I'm trying to understand meaning and application of the Maurer Cartan Form, but I'm still not quite there. I'm then trying to do some examples and trying understand how it works. I begun with the ...
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### The dimension of a projectivised, complexified tangent bundle

I am looking at page 15 here, starting from the line 'Let us suppose we are given a...' The Zoll projective structure is not relevant to this question, so don't worry about that. We take, in this ...
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### Why are Lie groups automatically analytic manifolds?

In the book by Kolar, Michor, and Slovak, it is shown that multiplication $\mu:G\times G\to G$ is analytic in some neighborhood of $e$. Specifically, they show that in the chart given by $\exp^{-1}$, ...
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### Elementary question: local computation of curvature on principal bundle

Let $G$ be a Lie group and $S=[0,1]^2$. Let $\omega$ be a connection $1$-form on the trivial principal $G-$bundle $P=S\times G$ over $S$. Let $(x_1,x_2)$ be coordinates on the base $S$. We can choose ...
Suppose $U \subset \mathbb{R}^n$ is open and $\mathbf{f}: U \rightarrow \mathbb{R}^m$ is $C^1$ with $\mathbf{f}(\mathbf{a}) = \mathbf{0}$, and $\mathrm{rank}(D\mathbf{f}(\mathbf{a})) = m$. Show ...