# 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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### A proof of exactness of closed 1-forms on the two-sphere

Remark. I'm aware that here the same problem is solved. I'm in trouble with the proof I'm going to quote. Consider the following Claim. Every closed 1-form $\beta$ on $S^2$ is exact. This is an ...
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### Boundary on a manifold

I was wondering how I can see if a manifold has a boundary just by looking at the surface? The thing is that I want to understand how to apply the Gauß Bonnet theorem to surfaces and there I need to ...
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### differential geometry : basic query about tensor notation and tensor products

I have a few very basic queries. I've been studying differential geometry as part of a course on General Relativity, so I don't have a very well grounded understanding of the mathematical formalism; ...
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### Understanding Lipschitz domain

Here is the definition of Lipschitz domain given by Wikipedia. Let n ∈ N, and let Ω be an open subset of Rn. Let ∂Ω denote the boundary of Ω. Then Ω is said to have Lipschitz boundary, and is called ...
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### Visualizing Ricci scalar curvature

I am trying to learn more about Ricci scalar curvature. I am trying to get an image in my head of what scalar curvature actually represents about the curvature of a manifold. The most familiar image I ...
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### What's the difference between a directional derivative and a derivation?

I asked my uncle what a derivation is and and he wrote the following: Most calculus courses discuss directional derivatives and include geometric applications to surfaces of the form $G(x,y,z)=0$,...
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### Is every $S^1$ knot orientable?

let $K$ be an $S^1$ knot, i.e., $K$ is a homeomorphic copy of the "standard" $S^1$ living in $\mathbb R^3$ ). I am trying to see if $K$ is orientable. This is what I have: 1) If the map sending $S^1$ ...
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### An integral with respect to the Haar measure on a unitary group

Let $A,D\in \mathbb{C}^{n \times n}$ be diagonal matrices. I need to calculate $$\int_{U(n)}\det{(A-HDH^\dagger)}\,\mathrm{d}H$$ where $dH$ is the unit invariant Haar measure on the group of unitary ...
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### Flat connection of a vector bundle over a 1 dim. manifold

I'd like to show that a connection of a vector bundle $E$ over a 1 dim. manifold $M$ is flat, or equiv. that its curvature is zero. Let $D$ denote the connection, $\sigma$ a section of $E$ and $v,w$ ...
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### Projection map from $\mathbb R^n-(0)$ to $\mathbb RP^{n-1}$ is smooth

How do I show that the projection map from $\mathbb R^n-(0)$ to $\mathbb RP^{n-1}$ taking $x$ to its equivalence class $[x]$ is smooth?
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### Tangent space of a Product of two manifolds

Suppose $M$ and $N$ are two $C^\infty$ manifolds. Take $p\in M$ and $q\in N$. We have the following maps between these: $\iota_1 : M\to M\times N$, $\iota_2:N\to M\times N$, $\pi_1:M\times N\to M$ and ...
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### References for mean curvature flow in Riemannian manifolds

I am interested in mean curvature flows (MCFs) in Riemannian manifolds. But most textbooks about MCF seem to treat mainly MCFs of hypersurfaces in $\mathbb{R}^{n+1}$. Should I study them before ...
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### How to find the points of self intersection of Cayley's Sextic?

I am given that $Y(t)=\cos^3(t)(\cos(3t),\sin(3t))$ and need to find the unique point of self intersection. So I assumed $$\cos^3(t)(\cos(3t),\sin(3t))= \cos^3(u)(\cos(3u),\sin(3u)).$$ I took lengths ...
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### Why do we require that a complex manifold has the structure of a real manifold?

I am taking a course in complex manifolds, heavily influenced by Huybrechts' book "Complex Geometry", and in it we define a complex manifold $X$ to be a smooth, real manifold $M$ together with an ...
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### Does parallel transporting require an ambient space?

Can someone summarize why an ambient space isn't needed to measure curvature when parallel transporting tangent vectors or vector fields along a curve on a Riemannian manifold? How do we define the ...
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### What's the name of a solid that results from extruding an area straight along an axis?

If you have any kind of 2D shape and move it up into the third dimension, what do you call it, because it seems like prism is used only if the base is a polygon. It also seems like extrusion is a ...
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### Proof in Hamilton: Divergence theorem for differential forms?

For a vector field $X\in\Gamma(TM)$ on a closed Riemannian manifold $(M,g)$ we have \begin{align*} \int_M\text{div}X\;\mu=0, \end{align*} where \begin{align*} \text{div}X=-g^{ij}g(\nabla_iX,\partial_j)...