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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72 views

a question about pre-symplectic manifold

Let $(M,\omega)$, is pre-symplectic. Then can we say, ker$ \omega$ is subbundle of tangent bundle $TM$?
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53 views

Geodesic Interpolation of a Vector

I have two vectors given and I want to estimate another vector by using geodesic interpolation, how can I do this? Thanks
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67 views

How to construct weak-star convergence?

From partial derivative as vector basis $\left\{\dfrac{\partial f}{\partial x_i}, i=1,\ldots,n\right\}$. How to contruct sequence $\{u_i\}, i=1,\ldots, n$ such that: 1, $u_i \stackrel{w^*}\rightarrow ...
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110 views

Who's doing stochastic calculus on manifolds in the Netherlands on a professional level?

This is not a direct math question but as a graduate student of mathematics at the VU Amsterdam I am looking out for Ph.D. positions on interesting math topics in the neighborhood. I was wondering if ...
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72 views

Definitions of Semisimple Lie Algebra

We usually define semisimplicity of a Lie algebra $\mathfrak{g}\subset M_n ({\bf R})$ from two ways. I want to know the relation between them. One of the definitions of semisimple Lie algebra is ...
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71 views

Prolate spheroidal coordinates

If $\alpha \in (0,\infty)$, $\beta \in (0,\pi)$ and $\theta \in (0,2\pi)$. $$\varphi (\alpha,\beta,\theta) =( \sinh(\alpha)\sin(\beta)\cos(\theta), \sinh(\alpha)\sin(\beta)\sin(\theta), ...
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51 views

How prove prove the identity $J_{t}=J\triangledown\cdot V$

prove the identity $$J_{t}=J\triangledown\cdot V$$ where $J=\begin{vmatrix} x_{u}&x_{v}\\ y_{u}&y_{v} \end{vmatrix}$ is the Jacobian of the map $x(u,v;t)\in R^2$ and $$V(x)=x_{t}$$
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189 views

Show a cone isn't a regular surface

I know that a cone isn't a regular surface because I can't construct a chart with cts partial derivatives at its tip. But can anyone show me this last step rigorously? Why would any chart for the tip ...
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57 views

Newman-Penrose tetrad questions

I have a question/exercise relevant to students of mathematical relativity: Let $\left \{ l^{a},n^{a},m^{a},\bar{m}^{a} \right \}$ be a Newman-Penrose tetrad, where only the direction of $l^{a}$ is ...
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72 views

Existence of Solution: Embedding from 2D Euclidean space to a circle

Given a real matrix $X$ with $n$ rows and 2 columns, can the matrix be transformed to a real matrix $Y$ such that all the points formed by the rows of $Y$ lie on a circle (2d) and their inter-point ...
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83 views

De Rham Cohomology of Product of Manifold with an Open Interval

Let $X$ be a submanifold of $\mathbb{R}.$ Prove that $H^{k}_{DR} (X) = H^{k}_{DR} (X\times (0,1)).$ I know that we should consider maps $\iota_a: X\to X\times (0,1)$ by $\iota_a(x) = (x,a)$ for ...
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283 views

the normal curvature for torus

Hello everyone I need little help in differential geometry , I need someone can solve this problem. Q1 The surface of torus given by $X(U,V)=((a+b\cdot cos(U))\cdot cos(V),(a+b\cdot ...
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32 views

Reference of spin structure

I am looking for some elementary books (may be introduction) about Spin structure in general, and Spin structure on Riemannian manifolds. Someone can help me? Thanks a lot!
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132 views

First fundamental form to find arc length and angle

If the first fundamental form of a surface is $I = du^2 + (u^2+a^2)dv^2$, find the arc length of each edge and each angle of the triangle enclosed by the curves C_1: u = (a/2)v2, C2: u= (-a/2)v2, C3: ...
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76 views

Why Local Minimum is calculated for a derivative function instead of actual function?

In Machine learning regression problem, why the local minimum is computed for a derivative function instead of the actual function? Example: http://en.wikipedia.org/wiki/Gradient_descent The ...
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84 views

A problem in differential geometry

How can we get $ \large w= \frac{1}{k(a)} + \frac{1}{k(a+pi)}$ by using those $4$ facts I got? Let $y (a)$ be a simple closed planar curve with curvature $k > 0$ parametrized by $a$, where $a$ is ...
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96 views

Existence of Spin Group

"In mathematics the spin group Spin(n) 1[2] is the double cover of the special orthogonal group SO(n), such that there exists a short exact sequence of Lie groups As a Lie group Spin(n) therefore ...
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68 views

Codimension one foliation

let $f:M \rightarrow \mathbb{R}$ be a nowhere vanishing function defined on a Riemannian manifold $(M,g)$. consider the distribution $\ker(df)$. This is clearly involutive and thus defines a ...
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238 views

Pullback bundle intuition (differential geometry)

Can someone give me an intuitive explanation of the pullback bundle of a vector bundle in differential geometry? You can apply it to the tangent bundle as that's probably easier to visualise. Am I ...
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74 views

geodesic and geometry

Is Geodesic a signature of Geometry of a Space and if it is then it should also have a co-ordinate independent definition perhaps should be incorporated in Tensor Mathematics with a associated name ...
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67 views

Surface of a 2-sphere expressed as union of two closed disks

I'm reading a First Course in Differential Geometry by Chuan-Chih Hsiung and on page 8 he says "A closed disk that is homeomorphic to $I^2$ [i.e. $I\times I$, where $I = [a, b]$] is connected. The ...
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26 views

How to view $V^*$ as the space of $C^{\infty}(M)$-linear functionals from $V$ to $C^{\infty}(M)$

Suppose $V$ is a vector bundle and $V^*$ is its dual bundle. I was told that we can view $V^*$ as the space of $C^{\infty}(M)$-linear functionals from $V$ to $C^{\infty}(M)$. Can anyone show me how? ...
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26 views

Why $C^{\infty}(M)$ module of sections of a vector bundle $E\rightarrow M$ is a reflexive module?

As the title saying, Why $C^{\infty}(M)$ module of sections of a vector bundle $E\rightarrow M$ is a reflexive module? Here we are considering vector bundles with finite-dimensional fibers.
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45 views

What are the lense spaces?

Is there an easy explanation and maybe visualization for the lense spaces? What is so important about them? What do we know about curvature of a lense space?
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109 views

Chain rules for differential forms

I have the variable $x,y,z$ possibly depending on each other i.e. on a smooth manifold. Using the theory of differential forms I can derive $\left(\frac{\partial x}{\partial y}\right)_z ...
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270 views

Confusion about local diffeomorphism can imply global diffeomorphism

By definition (http://en.wikipedia.org/wiki/Local_diffeomorphism), that $f$ is a local diffeomorphism of $M$ means for every point $p$ of $M$, there exists a neighborhood $U$ of $p$ such that $f(U)$ ...
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120 views

Why continuous paths implies smooth path on the manifold?

On the page 32 of Lee's book Manifolds and differential geometry, he writes: In the definition of path connectedness..., we used continuous paths, but it is not hard to show that if two points ...
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65 views

How do I get $da+a\wedge a$?

Define the exterior covariant derivative $$d_{\nabla}:C^{\infty}(M,E\otimes \wedge^{k}T^{*}M\rightarrow C^{\infty}(M,E\otimes\wedge^{k+1}T^{*}M)$$ to be $$d_{\nabla}(s\omega)=\nabla s\wedge \omega+s ...
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62 views

fixed point on a manifold

Suppose we have a Riemannian manifold $M$ with an open subset $U$ and a smooth map $\theta: U \to M$. If there is point $q\in U$ such that $\theta(q)=q$ prove that $d\theta_q=Id$ as a map ...
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185 views

Quotient map of the complex projective space

It's not to hard to see that the quotient map $\pi\colon \mathrm{C}^{n+1}\backslash \{0\} \to \mathbb CP^n$ is smooth and surjective. Does that imply that it is a submersion as well?
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223 views

Christoffel symbols

I found this equation $$\sum_{k,l,r,s} g_{ik}\frac{\partial g_{jl}}{\partial x^{r}}g^{rk}g^{sl}\frac{\partial}{\partial x^{s}}=2\sum_{s}\Gamma_{ij}^{s}\frac{\partial}{\partial x^{s}}$$ here g's are ...
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135 views

The index of zero of vector field is well defined

Let $X$ be a vector field on a manifold $M$ of dimension $m$ (compact, connected, oriented) with isolated zero $z$. Let $B$ be a ball around $z$ such that $X$ has no other zeroes in $B$. We define the ...
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315 views

Little question about chain rule

How can I prove the usual chain rule, but in the context of smooth manifolds? I mean: let $f$ and $g$ be two differentiable maps (from $M$ to $N$ and from $N$ to $P$, respectively) and where all are ...
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67 views

Isoperimetric Area Function

I'm reading the definition of isoperimetric function area and appeared the following notation: $Ł^{n}_{g}(\Omega)$, where n is dimension of Riemannian manifold $(M,g)$. What is the meaning of ...
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48 views

Integral transformation

I'm familiar with the transformation theorem in $\mathbb{R}^n$: given $\varphi : \Omega \rightarrow \mathbb{R}^n$ which is a diffeomorphism, $\Omega$ open, then $$\int_{\varphi(\Omega)} f(y) dy = ...
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52 views

Vector fields: A calculation

Let $V,W$ be vector field of $M$, a smooth manifold. Let $\sigma: T_pM\to T_p^*(M)$ be a linear map. Then whether following inequality is true?? $$\sigma(V_p(Wf))= \sigma(V_p)(Wf)\text{ for any ...
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82 views

uniqueness of asymptote in manifold

Question 1 Let $M$ be a complete, noncompact Riemannian manifold, a ray $\gamma:[0,\infty) \rightarrow M$ starting from $p$, and a point $x \in M$ such that the asymptote $\widetilde{\gamma}$ ...
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103 views

A $C^\infty$ map $M^n \to \mathbb R^n$ must have singular points if $M$ is compact

Can anyone give me an hint: If $M$ is a compact manifold of dimension $n$ and $f:M\rightarrow \mathbb{R}^n$ is $C^{\infty}$, then $f$ can not be everywhere nonsingular. Suppose $f$ is everywhere ...
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60 views

Perpendicular intersection manifolds.

Given a vector field $X$ of the form $f_1(x_1,x_2) \partial/ \partial_{x_1} + f_2(x_1,x_2) \partial/ \partial_{x_2}$ on a smooth 2-dimensional manifold $M \subset \mathbb{R}^2$ with a fixed point at ...
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89 views

generalized pythogorean theorem in tensor form

I looked up http://mathworld.wolfram.com/MetricTensor.html but it does not seem to provide me exact formula of generalized pythogorean theorem using metric tensor and tensor. So, can anyone tell me ...
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55 views

Relation with Jacobi fields in a small neighbourhood of some point in a complete manifold

I have the following question: Let $(M,g)$ be a complete Riemannian manifold with all sectional curvatures non-positive. Let $p \in M$ and consider the function $d(x)=dist_{g}(x,p)$ in a ...
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119 views

Derivative/Chain Rule (for MANLYfolds) Computation

Embarrasingly, I can't compute the following derivative. $dh(X)=\left.\frac{d}{dt}h(e^{XT}]\right|_{t=0}$, where $X$ resides in the lie algebra of $\rm SL(3,\Bbb C)$ [ie $\mathfrak{sl}(3,\Bbb C)$] ...
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72 views

How to show this equation (hypersurface, differential geometry, calculus)

Suppopse $X:\mathbb{R}\times (0,T) \to \mathbb{R}^2$ is a parametrisation of a smooth curve $\Gamma(t)$ with $X(a + 1, t) = X(a, t)$ for all $a$. Let $v:\mathbb{R}\times (0,T) \to \mathbb{R}$ be a ...
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87 views

Elementary doubt regarding a tangent vector

The book I am using defines a tangent vector to $\mathbb R^3 $ at a point $p$; $\ v_p $ as the line segment $\ p+v $ though both p and v are points in $\mathbb R^3 $. My question is since all points ...
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126 views

Given Poincare Polynomial find the manifold.

Suppose we have a polynomial, is it always the Poincare polynomial of some manifold? I guess the answer is no, but don't know any example. Even more, if we have a ring, is it the cohomology ring of ...
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65 views

Can an interpolant have arbitrarily small curvature?

Consider a set of points $P=\{p_1,p_2,\ldots,p_n\}$ in the plane. Define $x_1$ (resp. $x_2$) to be the minimal (resp. maximal) $x$ coordinate of the points in $P$. Now, let $c(P)$ be any curve, ...
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495 views

How do You Define “Coordinate Transformation”?

What is the exact definition of "Coordinate Transformation"? Issues to Consider in Relation to the Question: Let us first consider an orthogonal system. The dot product of the vectors along the ...
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90 views

The lambda invariant

Consider the strip $\{x+iy: -1\leq x < 1 , y>1/2\}$ in the complex upper half plane and let $\lambda$ be the usual $\Gamma(2)$-invariant modular function on the complex upper-half plane. ...
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126 views

homology Questions

I have some questions and would be infinitely grateful to you for your answers: 1- $f^{*}$ being the dual of $f_{*}$ so the degree (between top dimensional (co)homology groups) is the same for both ...
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31 views

Describing the shape of these level sets.

Given the function $f(x,y,z) = y$ defined on $T^{2} = \{(x,y,z) : (\sqrt{x^{2} + y^{2}} - R)^{2} + z^{2} = r^{2}\}$ - the torus with inner radius $r$ and outer radius $R$ satisfying $0 < r < R$ ...