# 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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### What's the definition of being zero to infinite order at some point?

I'm reading a paper and it says The differential form $\psi$ is zero to infinite order at all points of $F$.That is,all coefficients of $\psi$ are zero to infinite order at points of $F$.(Here $\psi$...
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### Discretizations of Differential, Geometric and Topological Notions

I have noticed a recurring theme in Graph Theory / Theoretical Computer Science (abbreviated GT and TCS throughout this post) in that notions typically belonging to differential calculus / geometry / ...
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### Questions on J. F. Nash's answer about his errors in the proof of embedding theorem

In the interview of John Nash taken by Christian Skau and Martin Gaussen, in EMS Newsletter, September, 2015 when asked Is it true, as rumours have it, that you started to work on the embedding ...
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### Geodesic Formula in terms of First Fundamental Form

I may simply be overwhelmed by all the terms in this question, but I am at a point where I feel stuck: Given a surface $X(u,v)$ with $u=u(t)$ and $v=v(t)$, and $F=0$, find a formula for the geodesic ...
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### Area form of $M^2 \subseteq \Bbb R^4$: does $(-1)^{i+j-1}\det_{i'j'}(n,\nu) = dx^i \wedge dx^j$?

This is a follow up question from this one. I'm asking separately since one way or another, that one is sort of answered. Now, we know that if $M^{n-1} \subseteq \Bbb R^n$, then the volume element $dM$...
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### Proving the ratio of curvature and torsion is constant.

This question has been asked slightly differently in a few different forums, but I wanted to discuss my approach and see if I was on the right track: Prove that if the tangent lines of a curve make a ...
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### Reference request : manifolds and transitive lie group actions

I would like to show that any manifold $M$ with a transitive ation from a Lie group $G$ is diffeomorphic to $G/H$ where $H$ is the stabilizer of an element in $M$. Do you know any reference where I ...
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### Geodesics, isometries and connections.

I am trying to prove with the differential definition that isometries preserve geodesics. That is: Let $c(t)$ be a geodesic in a $n$-dimensional semirriemannian manifold $M$ and $g$ an isometry of $M$ ...
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### Fig 1.8 on page 16 of Guillemin and Pollack's “differential topology”

For fig 1-8 on page 16, there is a sentence explaining why it is not a submanifold: "The trouble arises because the immersion is not one-to-one". I am quite confused because the definition according ...
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### Definition of complete integrals - existence of envelope?

Let suppose a partial differential equation: $$\Phi(x,y,z,\partial_x z,\partial_y z)=0\qquad (1)$$ In some books I have found the following definition: Let $\Lambda$ and $\Omega$ be two open subsets ...
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### Simple proof of the existence of lines in the hyperbolic space

Let $\mathbb{H}^n$ be the hyperbolic space defined as warped product: $$g_{\mathbb{H}^n} = dr^2 + \sinh(r)^2 g_{\mathbb{S}^{n-1}}.$$ What is the easiest way to show that there exist at least one ...
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### Useful Formulas Analysis on Manifold

This is just a reference request. Does anybody has or know a book with a short handed formulary for Calculus on Manifold. I could do it, but surely someone already have done it better than I would. ...
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### Velocity vector of a parametrized curve

In fig. 2.4, at the point of intersection of curve shouldn't we will be having two velocity vectors? How to handle that? Or we will be having only one velocity vector.If that is the case then why is ...
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### Curve with arc length have signed curvature k(s)>0?

Let $g:I \to \mathbb{R}^2$ be a curve such that for all $s \in I$, $\|g'(s)\|=1$ and $\kappa_g(s) \neq 0$, where $\kappa_g$ is the signed curvature of $g$. Is $\kappa_g(s) \gt 0$ for all $s \in I$? ...
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### What is the index of a vector field with positive divergence?

Let $v:\Bbb R^n\to\Bbb R^n$ be a smooth vector field with an isolated zero at $z\in\Bbb R^n$. Suppose that $(\operatorname{div}v)(z)>0$. Can we say anything about the index of $v$ at $z$? I know ...
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### Curvature Scalar in Riemannian Space

Suppose that Riklm=a(gilgkm-gimgkl ) on some four dimensional Riemannian space and a is a constant. Question: Show that for the curvature scalar we have R=-12a. What I know from calculating the ...
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### Is a bijective local homeomorphism a global homeomorphism. What about diffeomorphisms?

Is a bijective local homeomorphism a global homeomorphism? What about diffeomorphisms? I don't know if it's true this property, I'm not sure. If someone can prove it I would be very grateful, and if ...
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### Geodesic curvature of a curve in the hyperbolic plane

Consider the curve $\gamma$ given by $y=b$ in the upper half-plane equipped with the hyperbolic metric $$\dfrac{dx^2+dy^2}{y^2}$$ Calculate the geodesic curvature of $\gamma$. The problem I'm ...
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### Why are symplectic manifolds and Riemannian manifolds so different?

They seem superficially similar in some ways 1) both have a given two tensor on the tangent space 2) if we choose a hamiltonian function for the symplectic manifold then both give a canonical ...
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### On the zero in the fibre of a vector bundle.

Let $X$ be a differentiable manifold, let $\{U_i \mid i\in I\}$ be an open cover of $X$, let $\{g_{ij}:U_i\cap U_j\to\mathbb{R}^r\}$ be a set of differentiable maps satisfying the cocycle condition, ...
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### Is it always possible to always choose coordinates so that the curvature is locally zero?

I would have thought that this was completely possible as manifolds are so "soft" and the only problems would have been global ones (like Gauss Bonnett etc). But I've never seen the phrase "of course ...
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### Question in integral curve

Can anybody please help me by explaining why they have evaluated $x_1(t)$ and $x_2(t)$ at $0$. Last second expression where they found $/alpha(t)$.
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### Points on the curve

We have to find points on the curve $ax^2+ay^2+2 bxy=c$ (Where c>b>a ) whose distance from origin is minimum . I am not getting any start . I am able to just find that the curve would be hyperbola
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### Laplacian of a submanifold in an Euclidean space

let $M^n$ be a smooth an $n$-dimensional sub-manifold of a $\mathbb{R}^{m}$ ($n<m$). Denote by $\nabla^{M}$ and $\nabla^{\mathbb{R}^{m}}$ be the gradient of $M$ and $\mathbb{R}^{m}$ respectively. ...
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