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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How to prove the following function is a diffeomorphism.

I am trying to prove that n-closed unit ball is Manifold with boundary. I constructed a function as follows. (Using Stereographic projection idea) $N$ is the north pole i.e, $ (0, \dots, 1)$ $X$ be ...
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58 views

Sobolev maps between manifolds.

Let $M, N$ be smooth compact Riemannian manifolds. I have a reference that defines the $k$th Sobolev space of maps from $M$ to $N$, denoted $H^k(M, N)$, by saying that one only needs to check that ...
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41 views

A $\mathbb{Z}$-graded Lie superalgebra from a Lie algebra

Let $\mathfrak{h}$ be any $\mathbb{K}$-Lie algebra. We set $\mathfrak{g}_{-1}=\mathfrak{h}$ (as vector space), $\mathfrak{g}_0=\mathfrak{h}$ and $\mathfrak{g}_1=\mathbb{K}$ (or any one dimensional ...
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25 views

Enlargable manifolds

Does anyone knows why enlargeable manifolds have this name? (a compact riemannian $n-$manifold is enlargeable if for every $\epsilon>0$ there is an orientable riemannian covering space which admits ...
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51 views

Exact $b$-metric

I am currently reading Melrose's book "The Atiyah-Patodi-Singer Index Theorem", and I am somewhat stuck in the section where exact $b$-metrics are defined. Let me briefly recall the relevant ...
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42 views

A question on the description on pullback of a function in C.C.Pugh's real analysis book

The page 322 of "Real mathematical analysis" by C.C. Pugh contains the following description about pullback of a function $T:\mathbf{R}^n \to \mathbf{R}^m$. Dual to pushforward is the pullback ...
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73 views

Surface normal of parametrized surface (cylindrical coordinates)

I do not really understand how things are parametrized, since all the definitions I have found have been in terms of cartesian coordinates. Is it possible to parametrize a surface in cylindrical ...
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39 views

Reeb Vector Field: Actual Construction in (R^3, Std) Contact Structure, given Open Book

say we have the Standard Contact Structure on $\mathbb R^3 , (r,\theta,z)$/~; $(r,\theta,z)$ ~ $(r,\theta,z+1)$, given by $ker(dz+r^2d\theta)$ ;we have an open book decomposition in which the pages ...
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52 views

Rheotomic surfaces parameterization?

Are there parameterizations for rheotomic surfaces? Or, am I stuck with implicit formulas and marching cubes for plotting points? Are there special cases where the surfaces are parameterizable? Here ...
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32 views

straightening a general annulus to a round one

My question deals with an assertion one can find in some proofs of "On the dynamics of polynomial-like mappings" by Douady-Hubbard (example prop.5): Let $A$ be an annulus of finite modulus with $C^1$ ...
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75 views

Stratification of a smooth map

I am trying to do an exercise. Namely, find the Thom-Boardman stratification of the smooth map $f(x,y,a,b,c,d)=x^2y+y^3+a(x^2+y^2)+bx+cy$, where $a,b,c$ are parameters. As I have seen, this is also ...
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81 views

Inclusion mapping in conformal compactifications

The conformal compactification of the prototype Euclidean space $\mathbb R^n$ can be carried out by embedding $\mathbb R^n$ into $\mathbb R^{n+1,1}$ with Minkowski signature. One then considers the ...
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101 views

Analysis on Manifolds, Lipschitz Vector fields

Let (M,g) be a smooth compact Riemannian manifold, $\phi:M\rightarrow M$ and $L^n_x=D\phi^n_x:T_xM \rightarrow T_{x^n}M$. Let $(L^n_x)^*$ be its conjugate with respect to the inner product induced by ...
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43 views

Piecewise smooth paths as a groupoid

Let $G$ be a set of piecewise smooth paths on a smooth manifold $M$. Define the source, target, identity, inversion, and composition of such paths in an obvious way. Is $G$ a groupoid? I don't assume ...
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232 views

Parallel transport on the 2-sphere

I would like to determine the parallel transport on the sphere $\mathbb{S}^{2} \subset \mathbb{R}^{3}$. Let $p_{0} \in \mathbb{S}^{2}$ and $\xi_{0} \in T_{p_{0}} \mathbb{S}^{2}$ a tangent vector to ...
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38 views

Surface Reconstruction from Hessian Field

I am looking for references regarding surface reconstruction. Consider a point cloud in $\mathbb{R}^3$ with the Hessian (or possibly second fundamental form) defined at each point. I would like to ...
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80 views

How to define degree of principal bundle

My question is basically the title: how does one define the degree of a principal G-bundle, for some group G? Is there an inherent notion of degree? Or does it depend on taking an associated vector ...
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52 views

Points positioned on a surface with maximum distance

Given a spherical shell with area A. I want to arrange n points on this surface in such a way, that the distance between those n points is maximal. Do you know how to do this?(Can we say something ...
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40 views

Optimization and Duality: Maximizing surface / Minimizing Boundaries

I have recently gone trough an optimization course. I just thought about common problems I encountered in many domains. Given the measure of a boundary (lenght in 2D, surface in 3D), what shape ...
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61 views

What is a formal definition of a paradromic ring?

Paradromic rings are surfaces (kind of like the möbius strip) that come of ataching one end of a (square paper) strip to the other forming a loop in such way that the (paper) strip has to twist around ...
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32 views

Extending a form trivially to a family

Let $\pi : X \to B$ be a family of compact Kahler manifolds over a smooth base $B$. We suppose that all the $X_b = \pi^{-1}(b)$ are smooth and that we can shrink $B$ as much as we like (so it can ...
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101 views

Compute multiple Rectangles area intersect by a circle

I've a need to compute the area of single elements (dice) of a matrix like this: http://i.stack.imgur.com/EKVSz.jpg The matrix is composed by 'c' columns and 'r' rows and every element/rectangle has ...
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56 views

Orthogonal Coordinate Systems Intuition

I'd really love it if you could give some intuition on how to derive the $x$, $y$ & $z$ coordinates from all/any of the orthogonal coordinate systems in this list, how you think about, say, ...
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40 views

isomorphism of two compex line bundles

I am looking for some non-trivial examples of Line Bundles and an example about isomorphism of two line bundles. With details
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37 views

Lifted Diffeomorphism

Suppose to have a diffeomorphism $\phi$ of the d-dimensional torus to itself, and suppose to lift it to a morphism of $\mathbb{R}^d$ to itself. I have proved that is still invertible. How i proof that ...
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89 views

Hamiltonian reduction for unit sphere

Let $(M, \omega)$, be Symplectic Vector space and $N\subset M$ be unit sphere. Then why $N/ker\omega\mid _N$ is naturally $\mathbb{P}^{n-1}(\mathbb{C})$ Here ker$\omega$=$\{ y\in M: \omega(x,y)=0, ...
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62 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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30 views

Parallelity of the pullback of 2nd fundamental form

I've got the following problem with the pullback of the second fundamentalform: Let $ \tau \rightarrow Gr $ be the tautological bundle over a Grassmannian (the fibre at a point is just the point ...
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45 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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65 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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100 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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70 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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66 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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71 views

Geodesic equation for a 2D manifold

I am having trouble understanding how the following statement (taken from some old notes) is true: For a 2D manifold such that $$ds^2=\frac{1}{u^2}(-du^2+dv^2)$$ If we assume that $$\dot x^a\dot ...
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73 views

Show that ${\alpha}$ is a line of curvature if and only if ${\alpha}$' is parallel to (Gradient of U in direction of (alpha)') along ${\alpha}$

A curve on M is a line of curvature if ${\alpha}$(t) is an eigenvector of the shape operator for all t. This is equivalent to saying that the unit tangent vector T(alpha) is a principal vector. ...
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24 views

The most general form of the metric for a homogeneous, isotropic and static space-time

What is the most general form of the metric for a homogeneous, isotropic and static space-time? For the first 2 criteria, the Robertson-Walker metric springs to mind. (I shall adopt the (-+++) ...
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42 views

Mymultiple image geometry

I have to work with multiple aerial images. the objective is to reconstruct 3d features. For a particular object, i want to find the images which are giving good viewing geometry than others. so ...
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27 views

Laplacian on Reductive coset spaces

Let us consider the sphere $S^n$ (embedded in $\mathbb{R}^{n+1}$) and let $X_i$ denote the vector fields which generate rotations about the $x_i$-axis. My questions are: (a) Is it true that ...
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50 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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172 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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50 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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77 views

Induced connections on induced bundles

Start with a connection on a vector bundle over a smooth manifold. This bundle has a morphism into a universal bundle with connection and any two such morphisms are smoothly homotopic. What is the ...
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20 views

Showing that $|e . \Theta (e)| \geq \min\{k_1,k_2\}$?

For the shape operator $\Theta :T_p(S) \to T_p(S)$ how would you show that if the principle curvatures of $S$ are non negative, $k_1, k_1 \geq 0$. Then for any tangential vector $e$ we have $|e . ...
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54 views

global vector fields in local coordinates

(Note that I am assuming that vector fields are just $\mathbf{R}$-linear derivations from $C^\infty(M) \to C^\infty(M)$, contrary to the popular definition). Given a coordinate patch $(U,\varphi)$, we ...
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70 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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74 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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262 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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30 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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126 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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74 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 ...