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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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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87 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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76 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 $C$ ...
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143 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 ...
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142 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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92 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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153 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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302 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 ...
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452 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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44 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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112 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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421 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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524 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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201 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. ...
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92 views

Tensors: summing over indices

Would anybody mind teaching me how to work these indices? Definitions: Throughout the following, repeated indices are to be summed over. Hodge dual of a p-form $X$: $$(*X)_{a_1...a_{n-p}}\equiv ...
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899 views

Hard Differential Equation. Please help.

first of all I'm not a mathematician, so I apologize if any of my understanding and terminology isn't up to par. Also, I've never used this website (or any of these kind of question/answer) websites ...
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129 views

Torsion of fundamental group is abelian

On Riemannian manifold with Ricci curvature bounded below (For example, flat torus. It has ${\bf Z}^2$ as fundamental group, which is nilpotent (=almost abelian). In fact Ricci curvature bouned ...
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95 views

Momentum map and equivariance

I am reading an article in which I do not understand some equivariance property about the momentum map. Let $G$ be a Lie group acting on a manifold $Q$. The action is denoted $(g,q) \, \mapsto \, q ...
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114 views

Asymptotics of Green's function of laplacian

Let $ \Omega $ be a domain in $ \mathbb{R}^2 $ with a Riemannian metric $ g $, and let $\Delta $ be the laplace-beltrami operator induced by the metric, with dirichlet boundary conditions. For $ x,y ...
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115 views

Parallel transport on a submanifold

Let $M$ be a Riemannian manifold and $N$ an embedded submanifold of $M$. Now I have a geodesic $c \colon (-a,a) \to N$ with $a>0$ and $c(0)=p$ in $N$, s.t. $c'(0)$ is orthogonal to $T_pN$. Is the ...
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222 views

Exact and Closed forms on Manifolds with Boundary

Let $\bar{M}$ be a manifold with boundary and let $M$ be its interior. Is this statement correct? A smooth k-form $\alpha$ on $\bar{M}$ is closed (exact) if and only if its restriction to $M$, i.e. ...
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32 views

Inequality in geodesic quadrupel

Let $(M,g)$ be a compact complete Riemannian manifold. Consider the geodesic quadrupel $ABCD$ where $l:=d(A,B)=d(B,C)=d(C,D)=d(A,D) \geq \frac{1}{2}$ and $d(B,D),d(A,C) \geq 1$. Let $x \in CD$ and $y ...
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83 views

angle sum for triangle on helicoid

Given the helicoid $$ \boldsymbol{r} = (u\sin v, u\cos v,v)$$ in three-dimensional Euclidean space, consider the triangle $T$ defined by $$ 0 \leq u \leq \sinh v, \qquad 0 \leq v \leq v_0.$$ The ...
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94 views

For submersion and submetry,why can we lift a geodesic “horizontally” to a geodesic?

A map $\sigma:X\to Y$ between locally compact complete inner metric spaces is called a submetry if $\sigma(B_r(p))=B_r(\sigma(p))$ for all $r>0$ and $p\in X$. Why is that a geodesic in $Y$ can be ...
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156 views

Commutativity of two vector fields. Prooving that $\frac{d}{dt}\mid_{t=0}\alpha (t)=[X,Y]_{p}$

If $X$ and $Y$ are smooth vector fields with flows $\phi^{X}$ and $\phi^{Y}$, then starting at some $p\in M$, if we flow with $X$ for time $\sqrt t$, then flow with $Y$ for time $\sqrt t$ and then ...
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50 views

Commutator of two “special” conformal Killing fields

Theorem 1.7 of Schottenloher's "Mathematical Intro to CFT" says that every conformal Killing field on a connected open subset $M$ of $\mathbb{R}^{p,q}$ for $n=p+q>2$ is of the form $$X^{\mu}(x) = 2 ...
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105 views

On differential geometry in Hilbert spaces

Suppose that $H$ is a Hilbert space and $M\subset H$ is a closed subset with non-empty interior and smooth boundary, whatever smooth boundary could mean. I wonder if the normal vector is onto on the ...
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255 views

Manifolds: A definition of the Gradient an Algebraic Tangent vector over charts, how to show equivalence?

Let X be an n-dimensional differentiable manifold and $p \in X$ . Let $(U, h, V )$ for X around p with coordinates $(x_1 , . . . , x_n )$ in V , and let $v_i , i = 1, . . . , n$ , be the basis of ...
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207 views

Chern class of tautological line bundle

I'm studying characteristic classes from the Chern-Weil construction (via connection and curvature). I'm trying to compute some simple examples. Let $E$ be the tautological line bundle over projective ...
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98 views

An orbit of a group action and the implicit function theorem

Suppose that a Lie group $G$ acts smoothly on a manifold $X$. We can easily prove the following theorem by using the constant rank theorem, which is a stronger theorem than the implicit function ...
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85 views

Fundamental Group, Piecewise Smooth Curves, Consevative Fields

Let M be compact Riemannian manifold. X be a vector field on M. I believe that work done by X any piecewise closed curve is zero, iff, the same is zero for a particular finite set of loops. I believe ...
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89 views

Following a polyline along the surface of a polygon that is twisted

I have (hopefully) an interesting problem regarding geometry. I will also search online and in literature but I thought to pose the question here as a third resource. For my problem I need to get the ...
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64 views

Stoke's theorem application to curl theorem. I did. Please can you check it?

Now, I need to apply stoke's theorem to curl theorem. My teacher gave a hint. Accourding to the hint, I accept $w=Pdy∧dz +Q dz∧dx + R dx∧dy$ $\in Ω^2(M)$ $dim(M)=2$ M is the subset of $\Bbb ...
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210 views

Show that the projection map is Orientation preserving iff n is even

My question is that Orient the unit sphere $S^n$ in $\Bbb R^{n+1}$ as the boundary of the closed unit ball. Let $U$ be the upper hemisphere $U =${$x∈S^n |x^{n+1} >0$}. It is a coordinate chart on ...
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51 views

Trivialization of a path of tamed almost complex structures

I am wondering if the following result is true: Let $(V,\omega)$ be a symplectic vector space and $\{J_t\}_{0\leq t\leq 1}$ a smooth path of almost complex structures on $V$ which are tamed by ...
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82 views

Pole of differential

Let $E : y^2 = x^3 + ax + b$ be an elliptic curve over the field $K$, char $K \ne 0$. We know that the differential $\omega = \frac{dx}{y}$ is holomorphic in infinity because we can write it as ...
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90 views

Curvature form projective spaces

Let $T\mathbb{C}P^n$ tangent bundle over $\mathbb{C}P^n$. We have an hermitian metric on $T \mathbb{C}P^n$ defined as $h=\frac{dzd\bar{z}}{(1+|z|^2)^2}$. If we consider Levi-Civita connection we can ...
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379 views

Chern classes tangent bundle.

I studied Chern classes but I don't find explicit examples of calculation. So I'd like to consider the tangent bundle over $\mathbb{C}P^n$ ($T\mathbb{C}P^n$) and develop the theory beginning from this ...
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130 views

Cotangent bundle of a complex projective space

How does the cotangent bundle of a complex projective space looks like? Is that an Einstein manifold?
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129 views

“Product” bundle notation.

Let $\newcommand{\Spin}{\operatorname{Spin}}M$ and $M'$ be two manifolds, equipped with a principal $\Spin_n$ and $\Spin_{n'}$ bundle called $P$ and $P'$, respectively. Then there is an induced ...
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74 views

How to define local sections on associated bundles?

On a $G$-principal bundle $P$ it's easy since $G$ has a preferred element, namely $e$ (identity), but there is no such element in arbitrary suitable space $F$ to construct local sections for the ...
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118 views

characterization of the differentiable functions over a regular surface

Let $S$ be a regular surface. And let $f:S\to \mathbb R$ be a differentiable function It's not hard to prove that if $ W$ is an open set of $\mathbb R^3$ such $ V\subset S\subset W$, and $f:W\mathbb ...
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58 views

Problem solution hint about boundary of boundary of chains from Arnold' book mathematical method

On his book Mathematical Methods of Classical Mechanics, (Chapter 7, Section 35, Problem 10), Arnold asks to show that the boundary of boundary of any chain is zero. He gives hint saying: by the ...
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281 views

Torsion of a closed curve

Why the total torsion of a closed curve on a sphere is zero? What is the meaning of total torsion? Is it mean that we should calculate the torsion from point A to point A (i.e. since the curve is ...
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84 views

Does the derivative of the derivative depend on a choice of connection?

Let $X,Y$ be smooth manifolds and consider the infinite-dimensional manifold $$ C^\infty(X,Y) $$ of smooths maps $f: X \to Y$. Note that there is an infinite-dimensional vector bundle $E$ over this ...
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116 views

Reason for defining Riemannian curvature tensor and torsion tensor in particular way

I saw how Riemannian curvature tensor and torsion tensor are defined, but I am not sure why they are defined that way. In 3-dimensional euclidean space with ordinary multivariable calculus, the ...