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 $\nabla_{X}:\Gamma(E) \to \Gamma(E)$ continuos aplication?

Be $\Gamma(E)$ smooth sections space of an vector bundle, $\Gamma(E)$ with Fréchet topology. Gived a conexion in E and a smooth vector field $X$, Is $\nabla_{X}: \Gamma(E) \to \Gamma(E)$ continuos ...
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44 views

What is the explanation of the equation 1-4.6 in the book “Applied Exterior Calculus”?

If the $n$ function $\{f^i(x^m)\}$ are of class $C^{\infty}$ in some neighborhood of $0$, then system of autonomous ODEs $$\frac{d\bar{x}^i(t)}{dt}=f^i(\bar{x}^m(t)),\quad\quad i = 1,\cdots, n$$ has a ...
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108 views

n-Torus with antipodal points identified

If we have n-torus $S^1 \times S^1 \times S^1 \times ....$ n times, and $\mathbb{Z}_2$ acts on this just sending each component of $S^1 \times S^1 \times S^1 \times ....$ to its antipodal. What will ...
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67 views

Geodesics of the Hyperbolic Plane.

Using the coordinates $\alpha=\log \frac{1+r}{1-r}$ and $\theta$ where $(r,\theta)$ are the usual polar coordinates, show that the segment of the y axis between $(0,0)$ and $(0,r)$ where $0<r<1$ ...
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23 views

Dual Translation plane

An affine plane $\mathcal{A}$ is called a translation plane if the translation group of $\mathcal{A}$ operates transitively on the point set of $\mathcal{A}$. So how do we define the dual translation ...
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106 views

Upper-half space of a manifold with boundary

Suppose that $X$ is a manifold with boundary and suppose that $x$ is a boundary point. Define the upper half space $H_x(X)$ in $T_x(X)$ to be the image of $\mathbb{H^k}$ under $d \phi_0:\mathbb{R^k} ...
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179 views

Geodesic question

Let $\mathcal{P}:=\mathcal{P}(\mathcal{X})$ be the $n$-dimensional manifold of all (strictly positive) probability distributions on $\mathcal{X}=\{x_0,\dots,x_n\}$. Each $p=(p(x_0),\dots,p(x_n))\in ...
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43 views

When is a delta function a valid distribution?

If $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is nicely behaved, one can view $\delta(f)$ as a distribution (linear functional on $C^{\infty}_c(\mathbb{R}^k)$)- but what if you don't have nicely behaved ...
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59 views

Does this Condition Force this Differential 2-form to be 0?

Let $dw$ be a 2-form; $\alpha$ be a 1-form, $g$ a function (i.e., a 0-form), $X$ be a fixed vector field , all living in a 3-manifold, so that $$dw(X,.)=g\alpha .$$ Does this condition force $dw$ to ...
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55 views

Connection on $\operatorname{Spin}^\mathbb{C}$ spinor bundle

Let $W$ be the canonical $\operatorname{Spin}^\mathbb{C}$ spinor bundle on a symplectic manifold $(M, \omega)$, with a compatible $J$ and $g$, so \begin{equation} {W_ + } = {T^{0,0}}{M^*} \oplus ...
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33 views

basis theorem in holomorphic tangent space

I know that if $(x^1, \cdots, x^n)$ is a local coordinate system in a manifold $M$ then $\{\frac{\partial}{\partial x^1},\cdots, \frac{\partial}{\partial x^n}\}$ forms a basis for the tangent space ...
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18 views

Last generalized curvature zero for a hyper-planar curve in $\mathbb{R}^n$?

Serret-Frenet frame has been generalized to curves in $\mathbb{R}^n$. I would like to enquire about what happens to the generalized curvatures if the curve belongs to one or more planes in ...
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82 views

Proof check for critical point definition with mean curvature

I'm currently trying to prove: "Definition: We say that a surface $S \in R^3$ is minimal if it is a critical point for the area functional" Starting with this: "If we consider a family of smooth ...
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38 views

Short examples that are/are not quantum-ergodic

Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic? Note that a (compact) Riemannian manifold is said to be quantum ergodic if ...
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98 views

Morse functions are dense in $C^{\infty}(M,\mathbb{R})$ questions.

Hi here is a proof inspired from the reference below. Feel free to get very technical with your comments so that at the end I understand it well. I am more concerned about the questions I added ...
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66 views

Manifold locally looks like a open set but not as a euclidean space?

I am reading about manifolds from the book by Millman. He says that manifolds locally looks like a open set, but there is no canonical way to make M look like a Euclidean Space, and so we can't define ...
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68 views

Does distance in hyperbolic space satisfy such properties which Euclidean distance have?

In Euclidean space $E^n$, the distance between two points $x, y$ is just $|x-y|$, and for each fixed $x_0$, the image $y\to\nabla_x|x_0-y|$ is $S^{n-1}$, so it satisfies ...
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41 views

How to compute saddle point index using sourcing flow lines?

Prove $Index_{p}(\bigtriangledown f)$= "dimension of sourcing flow lines from p" ,where p is a critical point. Attempt Near $ p \in Cr(f) $ in some coord. $ f(x) - f(p) = $ $\sum_j x_j^2 - \sum_k ...
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81 views

tubular neighborhoods - problem

Consider $\alpha:[0,L]\to \mathbb R^3$ a smooth regular simple closed curve, arc-length parametrized. Denote its trace by $\gamma$. Let $\epsilon >0$ be a real number and $N_\epsilon (\alpha)$ the ...
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131 views

Given Constant Ratio of Torsion to Curvature, Show Tangent times Constant Vector is Constant

Let $r(t)$ be a unit speed curve such that for all $t$, $\frac{\tau(t)}{\kappa(t)}=\cot(\theta)$ for some $0 < \theta < \pi$. Show that there is a constant vector $a$ satisfying $T(t) * a = ...
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41 views

PDEs along parametrized curves in $\mathbb R^2$

For Riemannian surfaces in $\mathbb R^n,n\geq 3$, the Laplace-Beltrami operator can be expressed in terms of the Riemannian metric and one can consider evolution equations (e.g. heat diffusion, wave ...
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23 views

How do I figure out the grid references necessary to create a circular 1 mile radius around one location?

How do I figure out the grid references necessary to create a circular 1 mile radius around one location? For instance: Grid Reference: SO 47904 87484 X (Eastings): 347904 Y (Northings): 287484 ...
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93 views

Surface of revolution

This problem is from Dubrovin's Modern Geometry (Problem 8, Exercise 8.4). Show that the only surfaces of revolution with zero mean curvature are the plane and the catenoid (which is the surface ...
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29 views

Help modeling 3d vector field

I'd like some help in finding the correct mathematical description of the stuff below: A sheet is deformed by a mass on it (like in one of those pictures showing the effects of General Relativity). ...
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278 views

Chain rule with covariant derivative

Let $\mathcal{M}$ be a $n$-dimensional Riemannian manifold with Levi-Civita connection $\nabla$. Consider the following function: $$\tilde{F}(v) = \operatorname{d exp}^{-1}_{p} ...
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48 views

Boundaries of a cycloid

Supposing the Catenoid is parametrized by $x(u,v)=(acosh(u)cos(v),acosh(u)sin(v),au)$,where $u$ is a real number and $0<=R<2\pi$ and $a>0$ is fixed. Given the pair of parallel circles ...
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64 views

Nonvanishing vector fields on $\mathbf{S}^2$ and bases for $T_p\mathbf{S}^2$.

A vector field on a manifold is a (continuous, differentiable) map $X: M \to TM$ such that $X(p) \in T_pM$ for each $p \in M$. The tangent space $T_pM$ has a basis. Take $M = \mathbf{S}^2$. For each ...
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42 views

Show that x defines a coordinate neighborhood on the upper sheet of the circular cone C

Consider the map $x : U = (0, 2π) \times (0,\infty)\to{\mathbb R}^3$, $(\theta, v) \mapsto (v\cos\theta, v\sin\theta, v).$ a) Show that $x$ defines a coordinate neighborhood on the upper sheet ...
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62 views

Is the $\varepsilon$-neighbourhood theorem used in proving Homotopic transverse extension?

In Guillemin & Pollack page 71 I can't see where "For compact mfld Y,the map $\pi:Y^{\varepsilon}\to Y$ is a submersion" is used to show: "If for $f:M\to N$, closed subset $C\subset M$, closed ...
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68 views

Curvature tensor on the sphere

While reading R. Hamilton paper "Three-manifolds with positive Ricci curvature" I came across the sentence: For the sphere we have $$R(u,v,u,v)=R_{ijkl}u^{i}v^{j}u^{k}v^{l}>0,$$ which is the ...
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64 views

Proving a curve of a logarithimic spiral is 1/s cot(theta)

A unit-speed plane curve $\gamma$ has the property that its tangent vector $t(s)$ makes a fixed angle $\theta$ with $\gamma(s)$ for all $s$. Show that: (i) If $\theta = 0$, then $\gamma$ is part of a ...
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34 views

If 2 Equations with different variables are equal, then they are constant?

Let $\phi:\mathbb R\rightarrow\mathbb R$, $\psi:\mathbb R\rightarrow\mathbb R$ be $C^{\infty}$ maps, and $f:\mathbb R^2\rightarrow\mathbb R^3$ by $$f(u,v):(u,v,\phi(u)+\psi(v))$$ and set ...
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54 views

Getting started with contact bundles

I'm currently reading William Burke's book Applied Differential Geometry and he uses a lot in the development of Lagrangian Mechanics the notion of a Contact Bundle. He does explain intuitively what ...
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68 views

Any material on complexification?

These days, I met a problem on linear algebra: Suppose $A,B$ are real matrices. If there's a complex unitary matrix $U$ such that $U^*AU=B$, where $U^*=\overline U^\top$, namely, the conjugate ...
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92 views

Consequences of tubular neighbourhood theorem

Consider an oriented manifold without boundary S embedded in an oriented manifold M. The tubular neighbourhood theorem says that there is a neighbourhood T of S in M which is diffeomorphic to ...
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57 views

Laplace-Beltrami operator for Kahler 2-form

Laplace-Beltrami operator for Kahler 2-form: $$\triangle\Omega(X,Y)=d\delta\Omega(X,Y)+\delta d\Omega(X,Y)$$ We know that $$\delta d\Omega(X,Y)=-\sum_{k}(\nabla_{e_{k}}d\Omega(e_{k},X,Y))$$ where ...
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52 views

Chains and cochains: integer versus real coefficients

Let a real, smooth manifold $M$ be given. For each non-negative integer $k$, let a singular $k$-cube on $M$ be a continuous mapping $c:[0,1]\to M$. Let $C_k(M,\mathbb Z)$ denote the set of formal ...
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50 views

Wedge product of a one- form and a Kähler form

Let $x$, $y$, $v$, $w$ be coordinates on $R^{4}$ and $g$ be the Riemannian metric whose matrix with respect to these coordinates is $$g=\left ( \begin{array} {cccc} 1 & 0 & -kx & 0\\ 0 ...
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58 views

Tanget space to manifold via curves without map

I define tangent space T to differentiable manifold, in point p, via equivalence class of curves. The condition for this equivalence is $(\varphi\circ\gamma_1)'(0)=(\varphi\circ\gamma_2)'(0)$ for some ...
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33 views

Proving a global property of the connection from a property of local connection matrices

Let $D$ be an $m$-dimensional distribution on an $n$-dimensional manifold $M$. Let $U\subset M$ be an arbitrary open subset such that on $U$ we can define vector fields $e_i$ such that for each $x\in ...
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50 views

Tensor vector bundle construction

$\newcommand{\p}{\partial}$Let $M$ be a smooth manifold, and define $$T_{r,s} := \bigsqcup_{p \in M} (T_p M)_{r,s} = \bigsqcup_{p \in M} \big( \underbrace{T_p M \otimes \dots \otimes T_p M}_r \otimes ...
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Prove that a canonical bundle is trivial

Consider a function $f \in C^{\infty}(\mathbb{R}^n)$, $y \in Reg(f), M=f^{-1}(y)$. Prove that the canonical bundle of M is trivial. I have an hint but I don't know how to use it: consider the open ...
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74 views

Structure equations on the 3-sphere

On the 3-sphere I have found the vector fields $X_1=(-x_2,x_1,-x_4,x_3)$, $X_2=(-x_3,x_4,x_1,-x_2)$, $X_3=(-x_4,-x_3,x_2,x_1)$, in the basis $\left\{\frac{\partial}{\partial ...
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88 views

How to prove that the composite function is smooth

Let $f:M \to N$ and $g:N \to K$ be smooth functions, where $M,N$ and $K$ are smooth manifolds. How to prove that the composite function $g \circ f$ is smooth, noting that the Chain Rule only applies ...
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69 views

construction of a special series of functions

Here is the problem: Let $A$ be the set of positive integers greater than 1. For each $L\in A$, we want to construct a smooth function $f_L$ with compact support such that ...
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46 views

Some properties in differential geometry of curves and surfaces

Let $\beta, \alpha$ be curves in $\mathbb{R}^3$ parametrized by arc length. Suppose $\beta$ is obtained by rotating $\alpha.$ Let $t_{\alpha}, n_{\alpha}, b_{\alpha}$ (resp. $t_{\beta}, n_{\beta}, ...
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26 views

Irreducible Decompositions of the Space of Sections of an Equivariant Vector Bundle

Let $G$ be a compact semi-simple Lie group, and $X = G/H$ a homogeneous space of $G$. If $V$ is a vector bundle over $X$, then is it true that $\Gamma^\infty(V)$ always has a decomposition into finite ...
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57 views

Vector fields on homogeneous space $G/H$

I am trying to understand why the vector fields on $G/H$ are maps $X:G\rightarrow \frak{g}/\frak{h}$ satisfying $X(rh)=Ad^{-1}(h)X(r),\,\, h\in H.$ Any hint would be greatly appreciated!
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162 views

Some Advice on My Undergraduate Paper

My teacher wants me to read something about "Differential Geometry in $R^3$" and choose a topic as a paper. Now I have finished these books. And I am interested in some topics below: $(1)$ ...
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55 views

Doubt on the definition of topological manifold

I need some clarifications on topological manifolds. My professor has defined them as second countable, connected, Hausdorff topological spaces which are locally homeomorphic to $\mathbb{R}^n_+$, ...