# 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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### 3-sphere complex co-ordinates

I am currently trying to understand some mathematical physics papers that deal with torus knots. I am trying to find the origin of a complex scalar field used. These fields are somehow related to the ...
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### Is the singular locus of a variety (as a variety itself) a smooth variety?

A general fact about the singular locus $Sing(X)$ of a variety $X$ (analytic or projective) is that they form a subvariety of the oringinal variety $X$. And we know that the boundary of a manifold ...
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### Does a differential manifold implies existence of unique tangent space at every point?

I like differential geometry and I want to know if a differentiable manifold implies unique tangent space at every point. I have searched but the definition I have found of differential manifold is ...
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### Does $T^3$ double cover $\mathbb{R}P^3$?

So $T^2$ double covers $S^2$, and $S^2$ double covers $\mathbb{R}P^2$. Therefore, $T^2$ quadruple covers $\mathbb{R}P^2$. I am looking at $\mathbb{R}P^3$ because I am interested in rotations. The ...
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### How to differentiate this equation?

I was trying to derive the catenary curve using the surface area of revolution equation then applying the Euler-Lagrange equation to minimize it and after some steps I get the equation of the catenary....
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Let $M$ be a smooth manifold. Let $d$ be any metric on $M$ which generates the topology of $M$. Let $f:M \to R$ be Lipschitz w.r.t the metric $d$. Is it true that $f$ is differentiable a.e? Note: ...
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### The second differential as a differential on the double tangent bundle

I know what the second differential of $f : \Bbb R^n \to \Bbb R$ means. Nevertheless, when working with abstract manifolds and in the absence of a connection, one cannot come up with a 2-covariant ...
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### First and second fundamental form with rotational surfaces (check)

I'm working out some examples for surfaces in differential geometry. I was working out simple rotational surface, but I think I've done something wrong. Let $\gamma\left(t\right)$ a curve ...
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### Product-like metric on a pseudo-Riemmanian manifold foliated by Lie group orbits

Suppose we have an $n$-dimensional pseudo-Riemmanian manifold $(M,g)$ on which a connected Lie group $G$ acts isometrically (I am most interested in the Lorentzian case if it matters). Suppose that ...
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### Is there a relationship between the pullback in differential geometry and that in category theory?

1. Is there a relationship between the pullback in differential geometry and the pullback in category theory? [2. Is there a relationship between the pushforward/pushout in differential geometry ...
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### Ham Sandwich Theorem - intuitive proof

Ham Sandwich Theorem. Given 3 measurable "objects" in $\mathbb{R}^3$, it is possible to divide all of them in half (with respect to their measure, i.e. volume) with a single 2-dimensional plane. Can ...
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### Conformal class of $\mathbb S^n$ [on hold]

What can we say about the conformal class of the sphere $\mathbb S^n$?
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### How do you find the metric tensor for a given manifold?

Is there some general way to derive the metric tensor for a given manifold M? For example, how was the metric for the surface of a sphere $$ds^2=d\theta^2+\sin^2\theta \, d\phi^2$$ first derived?
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### Total Gaussian curvature of a flat surface with cone singularities

Let $S$ be a surface and $g$ a riemannian metric on $S$ which is flat with finite number of isolated conical singularities of cone angle $\theta_i>2\pi$. I have two questions: 1) of course the ...
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### Can I choose this kind of neighborhood of a point on a curve?

$\textbf{Question}$ Suppose $f:[0,1]\rightarrow\mathbf{R}^{2}$ is a continuously differentiable, 1-1 function. If $f(a)\in f([0,1])$, then should there be some open ball $B(f(a),\epsilon)$, centered ...
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### What is the relationship between line of curvature and umbilical point?

I am guessing whether or not the following statement is true: All the points lie on a line of curvature of a connected curve are umbilical points. Conversely, given an umbilical point on a surface, ...
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### How to know whether a contact form is only defined locally or globally?

As described e.g. here the following is the standard contact form on $\mathbb R^{2n+1}$: $$\omega = dz + \sum_{k=1}^n x_k dy_k$$ Similarly, the following is the standard contact form on $S^{2n+1}$: ...
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### Convergence of a integral for every curve in the sphere

Let $S$ be the unit open sphere in $\mathbb{R}^3$: $x^2+y^2+z^2< 1$ and $\partial S$ its border $x^2+y^2+z^2= 1$. Let $f:S\cup \partial S\rightarrow \mathbb{R}$ be a continuous function which is ...
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### Proving this connection is actually the Levi-Civita connection.

Having a manifold $M$ and some vector bundle $E$ over $M$ I am familiar with the definition of a connection given by a function $\nabla:\chi(M)\times\Gamma(E)\rightarrow \Gamma(E)$ that satisfies the ...
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### Concrete examples and computations in differential geometry

I've been studying differential geometry by myself for some time now. I studied a fair amount of the basic general theory and gone through a lot of the exercises from several textbooks. Lately I ...
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### metric and homotopic maps on a manifold

Let $Y\subset \mathbb{R}^n$ be an embedded manifold without boundary. Prove that there is $\epsilon>0$ with the following property: If $f,g \colon X \rightarrow Y$ are smooth maps defined on a ...
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### Maximal offset distance for a surface

Let $\vec r = \vec r(u, v)$ be a regular (analytic) surface. Now we offsetting this surface to distance $d$ in normal direction; new surface is $\vec r' = \vec r + d\vec n$. New surface $\vec r'$ is ...
### function from a genus $2$ surface to $S^1$
Let $f\colon \Sigma \rightarrow S^1$ be a map from a genus $2$ surface to $S^1$. If $y\in S^1$ is a regular value of $f$ and $f^{-1}(y)$ is a nonseparating circle of $\Sigma$. How can I prove that $f$...