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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25 views

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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544 views

an injective immersion between two compact manifold of same dimension

$f:M\rightarrow N$ be a injective immersion, where $M$ and $N$ are same dimensional manifold with out boundary, we need to show $f$ is a covering map. what I tried is, $df_x:T_x(M)\rightarrow T_{f(x)}...
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60 views

Why do those terms vanish if the metric is Hermitian?

On this page, the author says: We now define a Hermitian manifold as a complex manifold where there is a preferred class of coordinate systems in which unmixed components of metric tensor vanish ($...
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44 views

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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48 views

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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34 views

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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18 views

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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33 views

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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44 views

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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30 views

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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24 views

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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39 views

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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The set of Morse point is open and dense, Exercise 3.3.23 in DoCarmo's Differential Geometry

Let $S\subseteq\mathbb{R}^3$ be a smooth surface, no boundary (not necessarily compact). For any $r\in\mathbb{R}^3\backslash S$, define a smooth function $h_r:S\to\mathbb{R},q\mapsto |q-r|$ Let $p\in ...
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35 views

If for each $\alpha, \beta$, the map $\psi_{\beta} \circ F \circ \phi_{\alpha}^{-1}$ is smooth, then $F$ is smooth.

Let $F:M\to N$ be a map. Suppose $\mathcal{A} = \{(U_{\alpha},\phi_{\alpha})\}$ and $\mathcal{B} = \{(V_{\beta},\psi_{\beta})\}$ are smooth atlas for $M$ and $N$ respectively. Suppose that for each $\...
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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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42 views

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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2answers
79 views

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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19 views

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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1answer
56 views

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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38 views

The graph of any function $f: \mathbb R^n \to \mathbb R$ is a level set for some function $F: \mathbb R^{n+1} \to \mathbb R$

Show that the graph of any function $f: \mathbb R^n \to \mathbb R$ is a level set for some function $F: \mathbb R^{n+1} \to \mathbb R$. Attempt: Define $F: \mathbb R^{n+1} \to \mathbb R$ as $F(x_1,...
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The Taylor expansion of the metric at the origin in geodesic coordinates

It is well known that in geodesic coordinates we have $$ g_{ij}=\delta_{ij}-\frac{1}{3}\sum_{k,l}R_{ijkl}x^{k}x^{l}+O(|x|^{3}) $$ I have been trying to find a rigorous proof of it, but I cannot find a ...
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2answers
124 views

Is $n$-dim manifold which is immersed in $\mathbb{R}^n$ diffeomorphic to a ball?

Let $M$ be an $n$-dimensional smooth compact oriented manifold with boundary which is immersed in $\mathbb{R}^n$ (codimension zero). Must $M$ be diffeomorphic to a ball with boundary (the closed unit ...
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Problems that differential geometry solves

Recently, I've been studying a course in differential geometry. Some keywords include (differentiable) manifold, atlas, (co)tangent space, vector field, integral curve, lie derivative, lie bracket, ...
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The isotropy of the action of $SU(3)$ on $\mathbb CP^2$

Consider the action of $SU(3)$ on the complex projective plane $\mathbb CP^2$. How we can find the isotropy group?
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What's the point of studying differential geometry? [duplicate]

I've been taking a graduate differential geometry course this semester, and since the beginning I have wondered why one should try to learn that subject. It doesn't mean I don't like it, because I ...
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curvature of planar hyperbola

If a planar hyperbola is parametrized as follows $$ x(t) = a\sec(t), \quad y(t) = b\tan(t), \quad a, b \text{ constants} $$ what is the curvature of the hyperbola curve?
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Avoiding the spherical polar coordinate singularity on $S^2$ by using a double cover?

Is it possible to avoid the spherical polar coordinate singularity on $S^2$ by taking the coordinates as they originally are on $T^2$, i.e. ranging from $0$ to $2\pi$ mod $2\pi$? How would one ...
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1answer
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$G$-structure of a product manifold

My question concerns $G$-structures on manifolds: Let $M$ be an $n$-dimensional manifold. Since any $n$-dimensional manifold admits a riemannian metric, $M$ admits an $O_n$-structure. Similarly, $\...
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49 views

$G$-structure defined by a tensor

Let $M$ be an $n$ dimensional manifold with its bundle of linear frames $\pi:L(M)\to M$. Suppose $T_0$ is a tensor on $\mathbb R^n$ and $u\in L(M)$. We may view $u$ as a linear map $u:\mathbb R^n\to ...
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Does a tensor quantity only depend on the vector field at a point?

In Riemannian geometry there are many functions that have vector fields as arguments. Some (like the curvature tensor) are tensorial and some (like $<\nabla_{X}Y,Z>$) are not. Does the value of ...
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2answers
75 views

Show that geodesic equation is given by $\ddot x^k +\Gamma_{ij}^k \dot x^i\dot x^j=0$

I know that $\gamma $ is a geodesic if and only if $$\nabla _{\dot \gamma}\dot\gamma =0.$$ Using this, I'm trying to re find the equation $$\ddot x^k +\Gamma_{i\ell}^k \dot x^i\dot x^\ell=0,$$ but I ...
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Almost complex manifolds are orientable

I want to verify the fact that every almost complex manifold is orientable. By definition, an almost complex manifold is an even-dimensional smooth manifold $M^{2n}$ with a complex structure, ...
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What is the smallest Euclidean space in which one can embed a given curved space?

Given a $d$-dimensional curved space, how many dimensions are required to embed it? As an example think of a sphere's surface, which is a two-dimensional curved space that can be expressed in ...
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1answer
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The isotropy of the complex projective plane for the action of $SU(3)$

If we consider the action of the compact real form $SU(3)$ of $SL(3,\mathbb C)$ on the space $\mathbb C^3$. Since the action is transitive, how to find the stabilizer $G_x$? Is it useful to find ...
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33 views

System of Stochastic Differential Equations (SDEs) from Diffusion on Manifold

I am looking at a system of SDEs due to Brownian motion on a 3d Riemannian manifold (see e.g. Ito, 1962, The Brownian Motion and Tensor Fields on Riemannian manifolds). I have reduced the associated ...
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What does it mean when a differential form “stays the same”?

For example, consider the differential one-form $$\frac{\mathrm dw}{1-w^2}$$ If we make the change of coordinates $w=1/z$ then we see that $$\frac{\mathrm dw}{1-w^2} \longrightarrow \frac{\mathrm dz}{...
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If $\pi:\widetilde{M}\to M$ is a covering map, $M$ is complete iff $\widetilde{M}$ is complete

Let $\widetilde{M}$ be a covering space of a Riemannian manifold $M$. Show that $\widetilde{M}$ has a Riemannian metric such that the covering map $\pi:\widetilde{M}\to M$ is a local isometry. Then $\...
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35 views

Understanding the notation $\nabla u \otimes \nabla u$

On a Riemannian manifold $(M,g)$ let $u = u(t,x)$ the solution to the heat equation $\partial_t u = \frac 12 \Delta u$. The Laplace-Beltrami operator etc. are taken with respect to the metric $g$. I'...
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Evaluating zero set $Z_{R}(f)$ as a line, the union of a line an a surface, …

The zero set is defined as: $$Z_{R}(f) = \{x \in \mathbb{R}^3 : f(x)=0\}$$ In this post its claimed that the zero set of $$f(x_1,x_2,x_3) = x_1^2 + x_2^2$$ is a line while the zero set of $$f(x_1,...
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For any smooth manifold, is it true that for any two points on the manifold, there exists a chart that covers the two points?

Some say that we can connect two points with a continuous curve and a small contractible neighborhood of the path together with the charts of the points can be regarded as the chart. But I don't know ...