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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is the image of a polynomial map contractible?

There are often questions in differential geometry asking if a certain manifold (say a circle) has a polynomial parametrization. Are there topological obstructions to existence of such parametrization?...
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Timelike Loop Spaces as Projective Null Twistor Spaces

I have already asked this question in physics.stackexchange, but have not got any answers, so I have decided to ask it here. Let $\mathcal{M}$ be a spacetime, and let $\Omega\mathcal{M}$ denote the ...
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195 views

Formula for integral over hypersurface??

Can someone give me a formula for an (Lebesgue) integral of a function $f:M \to \mathbb{R}$ where $M$ is a bounded $C^k$-hypersurface of dimension $(n-1)$ in $\mathbb{R}^n$? I have tried the ...
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186 views

How is Euler-Lagrange equation used to find optimal solutions in minimizing a function?

How is the Euler-Lagrange equation: $$ L_x(t,q(t),q'(t))-\dfrac{d}{dt}L_v(t,q(t),q'(t))=0 $$ used mathematically in finding the optimal solutions of minimising a function? Can someone give me an ...
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45 views

Difference between diffeomorphisms fixing a point or a whole neighborhood.

Let $S_g$ be a closed orientable surface of genus $g$ and $S_{g}^1$ a closed orientable surface with one boundary component. Let $p$ be in $S_g$ and let's note $\mathrm{Diff}_+(S_g,p)$ the set of ...
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67 views

Inverse Function Theorem (results using it)

Hi i'm thinking in some ideas for my bachelor thesis. I'm working in a more "general" framework than manifolds, and i found that the Inverse Function Theorem is valid in such structures. So i was ...
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95 views

Geometric Cauchy Problem

I'm attending a course in Symplectic Mechanics and I have some problems in understanding something written in my lecture notes. We are in the following setting: let $Q$ be a manifold (of dimension $n$...
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52 views

Most natural symplectic structure?

Suppose I have 2-dimensional manifold embedded in $\mathbb{R}^3$. It's clear that the most natural Riemannian metric is the one induced by the usual inner product. What about symplectic forms? Is ...
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62 views

Restriction of a Lie bracket on the space of section of a vector bundle..

Let $A\longrightarrow M$ be a vector bundle and $U\subseteq M$ an open set. Suppose I have a lie bracket on $\Gamma(A)$ such that if $\rho:A\longrightarrow TM$ is a bundle map then $$[a, fb]=f[a, b]+\...
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78 views

Can anyone explain how to calculate Gaussian and mean curvatures using B-Spline fit method?

I want to calculate Gaussian and mean curvatures of some real surface which is from some RGB-D camera like Kinect. So there is no a expression of this real surface. How to calculate the Gaussian and ...
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238 views

Volume form on $\mathbb{S}^2$

Let $\omega = x_1 \,dx_2 \wedge dx_3 + x_2 \,dx_3 \wedge dx_1 + x_3\, dx_1 \wedge dx_2$, with $(x_1,x_2,x_3) \in \mathbb{S}^2$, be a volume form on $\mathbb{S}^2$, and let $f : \mathbb{S}^1 \times ]-\...
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472 views

Prove that circle is the only curve which spherical indicatrix coincides with it

The task is to prove that a space curve and its spherical indicatrix of tangents coincide if and only if the curve is a circle. Def. As a point moves along a space curve C envision a unit vector t ...
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46 views

FLRW metrics (isotropic and homogeneous space)

Consider a spacetime with metric $$ ds^2 = -dt^2 + a^2(t)d\Omega_k^2, \quad k=0,\pm1$$ where $a(t)$ is any regular function and $d\Omega_k^2$ is the 3-dimensional metric of the 3-sphere $S^3$, if $...
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64 views

Help with local coordinates computations-differential forms

I am reading a lecture notes on differential forms and tangent bundles on complex manifolds and I got stuck on one line where the author does not explain how he did the computation: Let $\...
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95 views

Definition of contact metric structure

I know this is a rather stupid question, but I still need to ask (and I am a physics student, so please excuse me using components): In Blair's book and many other litereatures, the definition of a ...
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47 views

On the Geroch's argument

During the study of Geroch's argument to prove positive mass theorem, I faced a problem explained below: Suppose $(M,g_{\mu \nu})$ is a four dimensional Lorentzian Manifold and $\Sigma$ is a ...
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190 views

norm of tangent to geodesic is constant

How do you prove that $g(T, T)$ is constant along a geodesic, where $g$ is a metric and $T$ is the tangent vector to the geodesic?
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148 views

Definition of a tangent space

Today we defined a tangent space similar to the description here: enter link description here My problem is the following: Why do we need to refer to charts in this case? I mean, would it not be ...
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173 views

Relation between exponential map and parallel transport

I'm starting to learn Riemannian geometry and have a question. Let $\mathcal{M}$ be a Riemannian manifold, $p \in \mathcal{M};\ \tau_{p}^{q}$ be a parallel transport from $p$ to $q$ and $\...
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85 views

Why are geodesics preserved by the quotient with the isometry group $M/G$?

I'm trying to prove that if $\langle M,g\rangle$ is a riemannian manifold and $G = Isom(M)$ acts properly discontinuous on $M$, then a geodesic $c$ is send to another geodesic by the map $\pi: M \...
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72 views

Show that the Mod 2 Linking Number is well defined.

If $X$ and $Y$ are compact, boundaryless smooth manifolds, $f\colon X \to \mathbb R^n$ and $g\colon Y \to \mathbb R^n$ with $f(X) \cap g(Y) = \emptyset$, and with $\dim (X) +\dim (Y) =n-1$. Define ...
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47 views

show that $g\circ f :S_1 \to S_3$ be also smooth and $d_p(g\circ f)=(d_{f(p)}g)\circ (d_pf)$

proposition: Let $S_1$, $S_2$, $S_3$ be 3-regular surfaces and $f:S_1 \to S_2$ and $g: S_2 \to S_3$ be smooth maps. Then show that $g\circ f :S_1 \to S_3$ be also smooth and $$d_p(g\circ f)=(d_{f(p)}...
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205 views

$k$-jets of sections of a vector bundle..

I need some help for establishing a connection between two definitions of $k$-jets: Algebraic Definition: Let $E\rightarrow M$ be a smooth vector bundle and define the ideal of $\Gamma(E)$: $$I_p(M):...
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58 views

Global conformal equivalence of two regions of Minkowski spacetime

I am wondering whether the region $H:=\{(t,x) : x^2-t^2<1\}$ of $(1+1)$-dimensional Minkowski spacetime, equipped with the restriction $g_H$ of the standard Minkowski metric $g=-\mathrm{d}t \otimes ...
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45 views

Comparing the norm of a trace of a curvature tensor with the full norm

Let $V$ and $E$ be complex vector spaces of dimensions $n$ and $r$, equipped with hermitian inner products $\omega$ and $h$ respectively. Let $R$ be a curvature-type tensor, that is an element of $\...
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145 views

Vector fields on smooth manifolds and Lie algebras

I'm currently studying differential geometry on smooth manifolds using differential forms and I'm trying to apply it to what I have learned earlier about Lie groups, but something doesn't seem to ...
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182 views

Local trivializations for orthonormal frame bundle

Let $(E,\pi, M)$ be a real vector bundle of Rank $N$. Then one can define its frame bundle $GL(E)$ as follows: $GL(E)_x:=\{\text{ordered bases of }E_x\}$ (for $x\in M$). $GL(E):=\bigcup_{x\in M} GL(...
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64 views

Group generated by several vector fields

I have two (or more) smooth and integrable vector fields $v,w$ on a smooth manifold $M$. Each generates a flow map $\Phi_v$,$\Phi_w$ that forms a single parameter Lie group of diffeomorphisms. Let's ...
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167 views

Whitney Embedding theorem for manifold with boundary in Lee's Introduction to smooth manifolds(2nd Edition)

I am reading through the new edition of Lee's book and I am stuck by the proof of Theorem 6.15. When passing to a non-compact manifold, the author begin by defining several sub-levelsets and claim ...
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326 views

$\alpha \wedge \beta = 0$ iff $\beta = \alpha \wedge \gamma$

I have been given the following problem: Let $\alpha$ be a nowhere-zero 1-form. Prove that for a (p+1)-form $\beta$ $(p\geq0)$, one has $\alpha \wedge \beta = 0$ if and only if $\beta = \alpha \...
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59 views

Continuity of point wise orientation

Prove that $[(X_1,...,X_n)]$ on a manifold $M$ is continuous if and only if every point $p$ in $M$ has a coordinate neighborhood $(U,\phi) = (U,x^1,...,x^n)$ such that for all $ q \in U$, the ...
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605 views

plane curves and osculating plane

Let $\alpha$ be a curve such that $|\alpha'(s)|=1$ for all $t$ and $k\neq 0$. The tangent vector $\vec T(s)$ and the normal vector $\vec N(s)$ through $\alpha(s)$ span a plane called the osculating ...
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60 views

Finding an isometry given the distance between points

If $A,B,C,D \in \mathbb{R}^n$, such that $d(A,B)=d(C,D)$, then exists an isometry $f:\mathbb{R}^n\rightarrow \mathbb{R}^n $, such that $f(A)=C, f(B)=D$. Thanks a lot for the help.
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91 views

Metric on the sphere involving tensor product

The metric on the sphere in the $(\theta,\phi)$ is of the form: $$ g_{\theta\theta}=r^2,g_{\theta\phi}=g_{\phi\theta}=0,g_{\phi\phi}=r^2\sin^2\theta $$ When transforming it to the $(x,y)$ coordinate ...
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77 views

minimise the total distance to a hyperbolic curve from two fixed points

The point $C$ moves along the hyperbolic curve which is given by $\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z^2}{c^2}=1$. The distances $d_{0}$ and $d_{1}$ in from $A$ to $C$ and $B$ to $C$ respectively. ...
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150 views

Smooth boundary is a smooth manifold

Let $M\subset \mathbb{R}^n$ be a $k$-dimensional manifold and $X\subset M$ a subset. The boundary of $X$ in $M$, denoted by $\partial_M X$, is the set of all $x\in X$ such that each neighborhoud of $x$...
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145 views

Legendrian Isotopy of Knots can be extended to an ambient Contact Isotopy

I am attempting to understand a proof that an isotopy of two Legendrian knots $L_0$ and $L_1$ in a closed contact manifold (M,$\xi$) can be extended to an contact isotopy $\phi$ of M such that $\phi_0 ...
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95 views

Why a regular surface could not have boundaries?

I'm reading the differential geometry written by DoCarmo and having trouble when understanding the definition of regular surface. What troubles me is that I could not see why the definition would rule ...
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156 views

Maximal submanifold

I wonder if the notion of "maximal submanifold" exists or is relevant? I'm surprised because I found pretty much nothing about it on the web (after a quick search). The definition, which seems ...
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311 views

Construction of line bundles on the flag variety

Edit: The following is phrased in terms of algebraic geometry, but can be thought of analytically as well. Hence I added some tags... I am a bit confused about the subject in the title. For ...
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51 views

Transforming the measure in $CP^1$ mapping from Riemann sphere to $\mathbb{C}^2$-plane

I would like to know how the measure changes in $CP^1$ mapping from Riemann sphere (2-sphere) to $\mathbb{C}^2$-plane. Let a point on the 2-sphere is given by the vector $\hat{n}(\mathbf{x})=(n^x(\...
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464 views

pullback and pushforward examples

Where can I find some simple examples of pullbacks and pushforwards between manifolds. Specifically examples that show the details of the computations.
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45 views

Sufficient statistics and isometries

Let $(M,g)$ be an infinite dimensional statistical manifold with the Fisher information metric $g$. Is it true that any isometry on this manifold must correspond to a sufficient statistic?
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117 views

Problem about finding Gaussian curvature

The problem is about Gaussian Curvature When we define $M=r(u,v)$ be the surface parametrized by $(u,v)$ in $\mathbb{R}^3$ and $N$ be a unit normal vector of $M$. In my lecture we denote Gaussian ...
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434 views

Horizontal and vertical tangent space of Orthogonal group

We know for the orthogonal group n-by-n orthogonal matrices, the tangents are given by $X^T\Delta + \Delta^TX = 0$ where $\Delta$ is the tangent. Now I was reading about the vertical and horizontal ...
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530 views

local parametrization of regular surface

I am doing excercises of Do Carmo's dg of curves and surfaces Chapter 2.2 and need some help with the following excercise: Show that the set $S=\{(x,y,z)\in R^3;z=x^2-y^2\}$ a regular surface and ...
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208 views

Is this enough to show that this map has constant rank?

My question is related to this question I am given a $C^r$ manifold $M$ and a connected subset $A$ of $M$, and a $C^r$retraction $f:M\rightarrow A$ such that $f\vert_A=id:A\rightarrow A$. Then $A$ is ...
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81 views

Vector field from Lamination.

Let $S$ be a smooth closed (i.e. compact without boundary) surface. A geodesic lamination on $S$ is a nonempty closed subset of $S$ which is a disjoint union of geodesics. Suppose $\alpha$ is a ...
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62 views

Problem reduced to analyzing solutions of a family of nonlinear systems of equations

This was posted on mathoverflow about two weeks ago and I got no response so I'm asking here in case anyone has any ideas. Original post is here. I was able to reduce a research problem relating to ...
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125 views

$\deg(f)\neq 0\Rightarrow f$ is surjective

How can I prove the following statement? Let $f:M\rightarrow N$ be a smooth map between closed, connected, oriented manifolds of the same dimension. If $\deg(f)\neq 0$, then $f$ is surjective. ...