# 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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### Construct a diffeomorphism $\psi: B_1 \to \epsilon\text{-neighborhood of } K$, where $K$ is a subset of a smooth manifold.

I'm currently working through a paper by Brezis on the topology of Sobolev spaces. Right now I am having trouble understanding the following note made by Brezis. Let $M$ be a compact and smooth ...
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### Basic question: Curvature transforms under Complexified Gauge Transformation

Let $E$ be a holomorphic vector bundle over a Riemann surface $M$ equipped with a Hermitian metric. Let $\nabla$ be the compatible connection on $E$ amd $g$ is a self adjoint complexified gauge ...
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### How to find the curve for which the part of the tangent cut off by the axes is bisected at the point of tangency? [closed]

Find the curve for which the part of the tangent cut off by the axes is bisected at the point of tangency.
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### Derivative of Exponential map on manifolds

I'm trying to compute the derivative of the map $f:\Sigma\times [0,\delta)\to M$ given by $$f(p,t)=\exp_p tN(p),$$ in $X\in T_p\Sigma$, where $(M^n,g)$ is a Riemannian manifold, $\Sigma\subset M$ a ...
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### Projective linear special group diffeomorphic to $S^1\times \mathbb{R}^2$

How can I prove that $\mathbb{P}SL_2(\mathbb{R})$ is diffeomorphic to $S^1\times \mathbb{R}^2$? I was thinking about embedding $S^1\subset \mathbb{C}$ as rotations and $\mathbb{R}^2$ as dilatations (...
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### Geodesics in geodesic balls

It is well-known that in a geodesic ball centered at $p$, the radial geodesic between $p$ and $q$ is the unique minimizing curve. I'm trying to follow the proof of this given in Cheeger & Ebin (...
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### Surface element area from constrains

Consider a surface in $\mathrm{R}^n$ defined by $m$ linear constrains: $$\sum_i c_{ki} x_i = 0$$ We assume that the $m\times n$ matrix $c_{ik}$ is full-rank. Then there exists a linear ...
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### Explicit isomorphism $H (E, E^0) \cong H_{cv} (E)$

Yo! I've been trying to understand better why $$H (D, D-\{x\}) \cong H_c (D)$$ for a disk $D$ and, more generally, why $$H (E, E^0) \cong H_{cv} (E)$$. In general, the first isomorphism can be seen as ...
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### Integral of solid angle of closed surface from the exterior

Jackson derives Gauss's Law for electrostatics by transforming the surface integral of the electric field due to a single point charge over a closed surface into the integral of the solid angle, ...
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### The set of Riemannian metrics on a submanifold of $\mathbb{R}^{n}$

Consider a subset $U \subset \mathbb{R}^{n}$. Clearly, $U$ can be considered as a smooth ($n$-dimensional) submanifold of $\mathbb{R}^{n}$. A Riemannian metric on $U$ is a smooth map $g$ which ...
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### Property of geodesic in surface of revolution in $R^3$ [on hold]

It is a question of my homework , I really don't know how to start it .
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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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### When is the universal cover of a Riemannian manifold complete?

Let $(M,g)$ be a connected Riemannian manifold which admits a universal cover $(\tilde{M}, \tilde{g})$, where $\tilde{g}$ is the Riemannian metric such that the covering is a Riemannian covering. I ...
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### About a proof concerning the relation of Lusternik-Schnirelman-category and cup length

In the proof of the relation between Lusternik-Schnirelman-category and Cup length (of de Rham Cohomology) for smooth manifolds from this note (theorem 2) the argument goes: Let the given manifold $M$ ...
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