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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A curve is contained in a hyperplane if and only if $\kappa_{n-1} = 0$

I'm trying to prove that a curve $\gamma$ in $\mathbb{R}^n$ is contained in a hyperplane if and only if $\kappa_{n-1} = 0$, where $$\kappa_{i} = \frac{\langle e_{i}',e_{i+1}\rangle}{\| \gamma' \|}$$ ...
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60 views

Connection on Submanifold using given connection on ambient manifold

This result features in Hicks' Notes on Differential Geometry book.The theorem states that given $C^\infty$ fields X and Y on a submanifold M, we have $$\bar D_X Y=D_X Y+V(X,Y)$$ where $\bar D$ is a ...
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65 views

how to determine the curve

I have to determined the curve which passes through the point $(1/2, \sqrt3 /2)$ and cuts to each member of the family of circles $x^2+y^2=a^2$ forming a angle of $60º$ My idea is to create a ...
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50 views

question about regular mapping in elementary differential geometry by Oneill

I am looking at Oneill elementary differential geometry section 4.2 Patch Computations. In example 2.4, parametrization of a surface of revolution, it says Suppose that $M$ is obtained by revolving ...
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31 views

group of automorphism of an $SU(2)$-bundle

Let $H$ be the Hopf bundle over $S^2$, consider the vector bundle $H^k \bigoplus H^{-k}$ over $S^2$ and extend it to $S^2 \times R^+$, this gives an $SU(2)$-bundle $E_k$. The claim is that the group ...
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82 views

Dupin's indicatrix and asymptotic direction of a surface?

Asymptotic direction at a point $p$ of a surface $S$ is defined to be the direction of $T_{p}(S)$ for which the normal curvature is zero. And Dupin's indicatrix at a point $p$ of surface $S$ is ...
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53 views

Sections on the Tautological Line bundle $E(\gamma_n)$..

I have a question about the tautological line bundle over $\mathbb R\mathbb P^n$. Recall, this bundle is that whose total space is $$E(\gamma_n):=\{([x], v)\in\mathbb R\mathbb P^n\times \mathbb ...
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Why is positivity of $g$ required for $(x,v) \mapsto (x,g(x)v)$ to be smooth?

An exercise in Guillemin and Pollock (1.8.2) assumes that $g$ is a smooth, everywhere positive function on a manifold $X$. The book assumes all manifolds are embedding into some ambient Euclidean ...
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47 views

How to compute the scalar curvature of this submanifolds?

Hello : I would like to know how to compute the scalar curvature of the following submanfolds $ H = \{ ( t , x,y,z) \in \mathbb{R}^4 \ : \ t = ax^2 + by^2 + cz^2 \} $ of $ \mathbb{R}^4 $ using the ...
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75 views

Foliations transverse to a fiber bundle

In the book Geometric Theory of Foliations by Camacho and Neto a foliation $\mathcal{F}$ is called transversal to a fiber bundle $\pi: E \rightarrow B$ with fiber F if a)For every p in E the leaf ...
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63 views

Show that the inverse function is not continous.

I am working with the following exercise: Let $U = \{ (u,v) \in \mathbb{R}^{2} | -\pi < u < \pi, 0 < v < 1\}$, define $X: U \rightarrow \mathbb{R}^{3}$ by $X(u,v) = (\sin u, \sin 2u, v)$ ...
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102 views

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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70 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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47 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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26 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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58 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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44 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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102 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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42 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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60 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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78 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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82 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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117 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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46 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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284 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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42 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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57 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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70 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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115 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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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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40 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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98 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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73 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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113 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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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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86 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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285 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 ...