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 can we visualize a tensor?

I would greatly appreciate it if someone could explain to me how to visualize a tensor in the analogous way that we visualize a vector. Thanks!
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36 views

If for each $\alpha, \beta$, the map $\psi_{\beta} \circ F \circ \phi_{\alpha}^{-1}$ is smooth, then $F$ is smooth.

Let $F:M\to N$ be a map. Suppose $\mathcal{A} = \{(U_{\alpha},\phi_{\alpha})\}$ and $\mathcal{B} = \{(V_{\beta},\psi_{\beta})\}$ are smooth atlas for $M$ and $N$ respectively. Suppose that for each $\...
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90 views

Why are symplectic manifolds and Riemannian manifolds so different?

They seem superficially similar in some ways 1) both have a given two tensor on the tangent space 2) if we choose a hamiltonian function for the symplectic manifold then both give a canonical ...
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56 views

Cohomology of the Riemann sphere

Let us note $\overline{\mathbb{C}}$ the Alexandroff compactification of $\mathbb{C}$ (i.e. the Riemann Sphere). I can prove that $$H^1(\overline{\mathbb{C}},\Omega_{\overline{\mathbb{C}}}) \simeq \...
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39 views

Hyperbolic isometries and finite order elements

I'm reading a paper and I'm uncertain about one of its claims. I was wondering if someone could clarify. Namely, it states that for a discrete subgroup of $\text{Isom } H^n$, the finite order elements ...
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52 views

Surjectivity of the exponential map on SO(2n)/U(n)

Let $M:=SO(2n)/U(n)$ the homogeneous space of all orthogonal almost-complex structures on $\mathbb{R}^{2n}$. When $n=2$, it is known that $M$ is just the 2-sphere. 1) On the 2-sphere, the ...
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26 views

Gradient of a function in different coordinates

Let $U\subset \mathbb{R}^3$ be an open subset endowed with a triple orthogonal coordinate system ($x^1,x^2,x^3$) and $f\in \mathcal{C}^\infty (U)$ be a smooth function. The vector field $\nabla f$ ...
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44 views

local isometry between $x(u_{0},v_{0})= (\sinh u\cos v,\sinh u \sin v, v)$ and a helicoid

Define: \begin{equation} x(u_{0},v_{0})= (\sinh u\cos v,\sinh u \sin v, v) \end{equation} I want to show that there is a local isometry between $x(\mathbb{R}^2)$ and a helicoid: \begin{equation} y(u_{...
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28 views

For a minimal surface $M$ under Mean Curvature Flow, can it evolve between minimal surfaces continually?

I don't know much about this subject at all; I'm only just getting into it. As it turns out, a physicist friend of mine asked me a formulation of the following: Suppose $M$ is a surface in $\mathbb{R}...
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48 views

Book on differential Geometry with application to General Relativity

Does anybody know of a good book on differential geometry that has applications to general relativity and also focuses on geometrical intuition? I need a book that is not as rigorous as one that is ...
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32 views

Orientability of differantiable manifold of orthogonal matrices

I want to find out if differentiable manifold of matrices $M=\{A_{(3\times3)}(\mathbb{R}): A^TA=16E\}\subset\mathbb{R}^{3\times3}=\mathbb{R}^9$ is orientable. It is only worth proving that orthogonal ...
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48 views

Why say “exists smooth structure” instead of “is a smooth manifold”?

As some of you know I started to learn differential geometry some weeks ago. Now I came across this theorem (you can find it in Lee's Intro to Smooth Manifolds but it is in many other books also: ...
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27 views

When is a stable domain in a minimal surface area minimizing?

A stable domain $D$ in a minimal surface $S\subset \mathbb{R}^3$ is a domain for which the area-functional $A(t):=\int_{S_t}dS_t$ has non-negative second derivative, i.e. $A''(0)\geq 0$, for all ...
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18 views

Finding braches of equilibria

Consider the system of two equations in three variables $(x,y,z)$: $$\left\{ \begin{array}{rl} x+y+z+x^7+y^7+z^9 &= 0\\ x-y+z+1-\cos z &=0 \end{array}\right.$$ The point $x^* = (0,0,0)^...
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21 views

Mean curvature submanifold

Consider $S^{N-1}$ the unit sphere and let us focus our attention on the cap $$ G=S^{N-1}\cap\{x_N>0\} $$ with boundary $\partial G= S^{N-2}\times\{0\}$: it is quite obvious to see that $G$ is a ...
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16 views

Necessary condition for local existence of solution for system of two first order PDEs

Let $f,g$ be two smooth functions on $\mathbb{R}^2$ and $U,V$ be two smooth vector fields on $\mathbb{R}^2$. What is the sufficient condition for local existence of a solution $\phi$ of equations \...
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25 views

Apply flow of $V$ to a segment of a curve, Do you get covariant derivative?

Apply flow of $V$ to a segment of a curve, look at the velocity of the resulting new curve, a perturbation of the original curve. How is the new velocity (tangent) related to the original one (right ...
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37 views

Riemannian metric as an operator

In the article http://www.sciencedirect.com/science/article/pii/0370269379905896 authors consider principal bundle $P(M, G)$ and then define induced metric $g$ on $\eta = Sp(A)/G$, where $Sp(A)$ - ...
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31 views

1-form that compute the area of parallelogram

Find a form on $\mathbb{R}^4$ that compute the area of parallelogram generate by any pair of vectors $\vec{a}, \vec{b}$ which are in the plane $\pi=\{\vec{x}\in\mathbb{R}^4| \vec{x}=\vec{p}+s\vec{u}+t\...
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28 views

real analytic functions on manifold

Let $M$ be a real analytic manifold of dimension $k$. Is it then always possible to find real analytic functions $f_1, \dots, f_k \colon M \to \mathbb{R}$, such that they are functionally independent ...
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33 views

On Atiyah-Singer Index theorem and Atiyah-Bott fixed point theorem.

I would like to understand the statement of Atiyah-Singer and Atiyah-Bott theorems enough to see lots and lots of applications they have. My background is pretty thin with some basic algebraic ...
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22 views

Regular values of $g(x,y)= x^2 - y^2$

I am doing some very introductory studying about manifolds. I wanted to check I was getting the right end of the stick through this example. Could anyone verify/correct my solution to the following ...
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34 views

geodesic flow is proper action

Good evening to everyone. I'm having a problem in the following setting: If I'm having a homogeneous manifold $M=G/K$, where $K \subset G$ is a closed subgroup, I can always find a $G$-invariant ...
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17 views

Integral curves of time dependent derivations

Question: Given smooth manifold $M$, with algebra of smooth functions deoted by $C(M)$ let $D_t$ be a time-dependent derivation of $C(M).$ Let $\hat{D}$ be a derivation of $C(M\times \mathbb{R})$ ...
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33 views

Defining a differential for quotients

Let $f \colon M \to N$ be a smooth map between smooth manifolds and $f$ being a surjective submersion. Assuming we have a proper Lie-group action $G$ on $M$, with only one orbit type and $G$ acts on $...
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49 views

Meaning of alternation in definition of wedge product

Spivak defines the wedge product as $\omega \wedge \nu = \frac{(k+l)!}{k!l!} \text{Alt}(\omega \otimes \nu)$ and I have been running into some conceptual issues here. The alternation is defined as $...
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35 views

Mobius strip as manifold and as a bundle over $S^1$

I am trying to construct the Mobius strip bundle onver $S^1$. I am stuck in how to define charts for the Mobius bundle. Once I make this, the problem will easily be solved. My attempt was: $$M = [0,...
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26 views

On a parallelizable manifold, is there always a frame satisfying $[X_i,X_j]=0$?

Let $M$ be a parallelizable manifold. Is there always a global frame $(X_i)$ such that $[X_i,X_j]=0$ for all $i,j$ ? If the answer is no, what kind of obstruction there is to find such a frame ? ...
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37 views

Two definitions of the first Chern class

there are two definitions of the first Chern class that I don't know how to relate - hints and references are both welcome. So, first approach: say that I have a complex vector bundle $E\to M$; I can ...
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24 views

Normal vector field for an immersion

I know that a hypersurface $M$ of a riemannian manifold $N$ is orientable iff there exists a globally defined unit normal vector field $\eta : M \to TM^{\perp} \subset TN|_M$. Does the same hold for ...
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28 views

On Automorphism on Space of Smooth Functions

Given a Operator $T$ (an Automorphism) on the subspace $X$ of Smooth functions on $\mathbb R ^n$, $\mathcal C^\infty(\mathbb R^n)$ $$ X=\{u \in \mathcal C^\infty (\mathbb R^n) \,|\, supp \,(u)\...
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9 views

Metric of the doubling i.e circular billard table

Let $D^2=B_1(0) \cup_{\partial B_1(0)} B_1(0)$ denote the doubling in $\mathbb{R}^2$ i.e. the metric space which one gets after gluing two closed balls of radius $1$ with their induced metric coming ...
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15 views

parameterization of surface of revolution?

parameterization of surface of revolution formed by revolving the $x=\cosh z$ around z axis , i thought the it as $$x=\cosh z \cos \theta ,y=\cosh z \sin \theta ,z=z$$ Hence the surface can be ...
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19 views

Showing that a mapping is an isometry

Can anyone please help with this question? I have tried substituting but I'm not sure if that's correct so think I am missing something. Let S denote the surface of revolution $(x, y, z) = (\cos{\...
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26 views

How do I show that the reparametrization of a pre-geodesic is pre-geodesic?

So this is probably a very silly question but I need to prove the reparametrization of a pre-geodesic is a pre-geodesic. The hint in my book says it's obvious, so I am clearly missing something.
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34 views

Proper patch in the differential geometry

I have a question that coincides with this question. Proving that every patch in a surface $M$ in $R^3$ is proper. Prove that if $\mathbf{y}:E\to M$ is a proper patch, then $\mathbf{y}$ carries ...
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15 views

Finding a zero of homogeneous parts of polynomials using zero of those polynomials

Let $f, g \in \mathbb{Q}[x_1, ..., x_n]$ be polynomials of degree $d$. Let $F$ and $G$ denote the degree $d$ portions of $f$ and $g$ respectively. Suppose there exists a non-singular point $\mathbf{...
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25 views

Proving the invariance of the mixed product under direct congruent transformations

Text: Differential Geometry, by Erwin Kreyszig Given a set of vectors: $$ \overrightarrow{a}=\langle a_1, a_2, a_3 \rangle $$ $$ \overrightarrow{b}=\langle b_1, b_2, b_3 \rangle $$ $$ \overrightarrow{...
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30 views

Can the hyperbolic orbifold 2*55 be smoothly and isometrically embedded in 3-space?

Grow a square in the hyperbolic plane until its vertex angles become $\pi/5$. Assuming that the constant Gaussian curvature of our hyperbolic plane is $-1$, the sides of the resulting hyperbolic ...
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45 views

Discretizations of Differential, Geometric and Topological Notions

I have noticed a recurring theme in Graph Theory / Theoretical Computer Science (abbreviated GT and TCS throughout this post) in that notions typically belonging to differential calculus / geometry / ...
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16 views

Apply “thickness” to a minimal surface

By definition a minimal surface has no volume. But my goal is to give the minimal surface some volume. Is there a mathematical way to do this? I found a thread in a forum of a math visualization tool ...
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31 views

The map $f:M\rightarrow N$ is differentiable, where $M,N$ are $m$-manifolds.

Suppose the map $f:M\rightarrow N$ is differentiable, where $M,N$ are $m$-manifolds with $M$ compact and $N$ connected. If the degree of $f$ is 1, then $f$ is surjective?
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62 views

what does it mean to have inner product of $S^2$ and $R^3$?

It may be that the title of my question is wrong but i am writing this question because i am struck while reading this paper Brownian motion on rotational group Where $^*\mathscr{f} $ is transpose of ...
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48 views

Zero gradient in $L^2(M)$

I'd like to show that for $u \in L^2(M)$, for M a compact, connected Riemannian manifold, if $\nabla_g u = 0$ (i.e $\forall X$ $C^{\infty}$- vector field on $M$, $\int_M u \hspace{1mm} \text{div}_g X =...
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29 views

Can the rank of harmonic map decrease far from the boundary?

This question is in some sense a continuation of this question, though it asks something weaker. Let $(M,g)$ be an $n$-dimensional, connected, compact Riemannian manifold with boundary. Assume we are ...
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19 views

Is this proof about the Lie brackets and flows correct as given?

In this post, Fredrik Meyer gives a proof to the following formula(Please see the conditions and the meaning of notations in the link): $\frac{d}{dt}|_{t=0} \alpha(t) = [X,Y](p)$, where $\alpha(...
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74 views

Explicit form of the exponential map

I am stuck by the following problem. Let $ g = 1/y^2 (dx \otimes dx + dy \otimes dy)$ be a Riemannian metric on the half-plane $ S = \mathbb{R}^2_{y > 0}$. What is the explicit form of the ...
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29 views

On lifts of a trajectory of a quadratic differential

Let $X$ be a Riemann surface of genus $g\ge 2$ and $q$ a holomorphic quadratic differential on $X$. The differential $q$ defines a flat metric with conical singularities on $X$: if $q=f(z)dz^2$ ...
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27 views

A space curve with non-vanishing curvature is planar iff its torsion is 0

Intuitively this is simple and to prove the backwards direction: $\tau = 0 \Rightarrow \mathbf{b'}=0 \Rightarrow \mathbf{b} $ is constant. Then letting $\gamma$ be the parametrisation of the curve ...
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24 views

a algebraic concept in reimannian geometry?

let $(M,g)$ is a $C^{\infty}$ manifold with product form of $M=R \times \Theta $ the for each element $\varepsilon (r,\theta )\in M$, the tangent space $T_{\varepsilon}M$ is natural isomorphic to $...