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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Show that S is homeomorphic to a Klein Bottle

I've been struggling quite a bit with this question. Any hints/help would be greatly appreciated! Consider the quotient S = R^2/G where G = Z^2 acts by (n, m) • (x, y) = ((−1)mx + n, y + m) on R^2 , ...
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Explain why the open mobius band is a smooth surface

Explain why the open mobius band is a smooth surface and find a homeomorphic copy of it inside the real projective space RP^2 and inside the Klein Bottle K
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1answer
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Covariant derivative as a connection on a vector bundle

In the Wikipedia article Connexion (vector bundle), such a connection is defined as a function $\Gamma(E) \to \Gamma(E\otimes T^*M)$ . Then the definition of a covariant derivative is given as a ...
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1answer
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Tangent space to lie group at identity.

I'm supposed to show that for a Lie group G, $T_{(e,e)}G\times G \simeq T_eG\oplus T_eG$ and that $T_{(e,e)}m$ is given by $(X,Y)\mapsto X+Y$. I'm having trouble proving this. I'm not exactly clear ...
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42 views

Mathematical aspects of General Relativity

I was wondering what mathematical subjects are used in the study of the theory of General Relativity (black holes ...) I assume mostly differential geometry, Riemann Geometry ... but is there also ...
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Geodesic curvature

I am looking at the definition for geodesic curvature $\kappa_g$ of a path $\gamma:[a,b]\rightarrow X \subset \mathbb{R}^3$ in a smooth surface $X$ with unit normal $\hat n$. $$\kappa_g (s) = ...
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1answer
9 views

The tangent space of the boundary of a manifold with boundary is a subspace of the tangent space

I was trying to understand the following sentence in some notes I am reading: Let $X$ be a manifold with boundary. At any point $p \in {\partial}X$ there is a canonical subspace $T_{p}({\partial}X) ...
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26 views

Proving curve devides a sphere into two equal-areas

let $\gamma$ be a closed geodesic without points of self-intersection on a closed convex surface. Prove that the spherical image of $\gamma$ divides a sphere into two parts with equal areas I ...
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set of all regular values

Let $M$ be a compact manifold and $f: M\longrightarrow \mathbb{R}$ be smooth. Show that the set of all regular values of $f$ is open. How can I prove it? Could someone help me?
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Proving for differentiable curve $\kappa_1=1+\kappa_2^2$

Let $\gamma$ be a differentiable and regular curve in $\mathbb{R}^3$ which satisfies $|\gamma|=1$. Prove that for every point $$\kappa_1=1+\kappa_2^2$$ where $\kappa_1$ is the curvature of $\gamma$ ...
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Adjoint representation of the isotropie group of a homogeneous space

I have difficulties seeing why is the following true: Let $G$ be a lie group and $H$ a closed subgroup, with $\tilde{g}$ and $\tilde{h}$ their lie algebras. The adjoint action of $g\in G$ is given by ...
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Loxodrome equation

On Mathworld web site, the loxodrome is defined in oblate spheroid coordinates (even though with a peculiar "minus" sign in z, see here http://mathworld.wolfram.com/SphericalSpiral.html) where they ...
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23 views

Fundamental solution of Laplacian on manifold

I'm looking for a reference for the result that there exists a fundamental solution for the Laplacian on a flat torus $$\Delta \Gamma(x-y) = \delta(x-y), \quad x,y \in \mathbb T^2.$$ and that, ...
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1answer
33 views

closed but not exact

I saw several times that $\frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2}dy$ is closed but not exact. Closed, is obvious but I can't prove non exactness, can one please help me ? My attempt, let $f\in ...
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38 views

Unnecessary Elements in the Tensor Product?

For vector spaces $U, V$ there exits a unique (up to isomorphism) vector space, denoted by $U \otimes V$, and a bilinear map $\eta : U \times V \to U \otimes V$ such that for every bilinear map $\xi : ...
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1answer
50 views

What exactly does “differential forms are coordinate free” mean?

Most introductory texts on differential forms praise their property of allowing for a "coordinate free formulation". What exactly does this mean? What would be a concrete example for which a ...
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1answer
29 views

Fundamental solution of heat equation on a compact Riemannian manifold

Let $(M,g)$ be a compact Riemannian manifold of $m$ dimensional. Then there exists a sequence $(\phi_i, \lambda_i)_{i\in\mathbb{N}}\subset C^\infty(M)\times\mathbb{R}_{\geq0}$ such that ...
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how to test whether cobordism exist between two manifold or two system of polynomials

from wiki Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. from book geometrisation of 3-manifolds ...
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how to calculate the RicciTensor for system of 3 polynomials equations in 3 variables to test whether is spherical [on hold]

how to calculate the RicciTensor for system of 3 polynomials equations in 3 variables i find examples using differential expression, how to do for system of polynomial equations Riemannian metric of ...
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41 views

Different Definitions of Tensor product, Halmos, Formal Sums, Universal Property

In the classic Finite-Dimensional Vector Spaces by P. Halmos he defines the Tensor product as The tensor product $U \otimes V$ of two finite-dimensional vector spaces $U$ and $V$ (over the same ...
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19 views

Unit Disk Regular Surface?

I am having trouble proving these two problems: 1) is $\{(x,y,z)\in \mathbb{R}|z=0, x^2+y^2\leq1\}$ a regular surface? I say no because the closed unit disk is a closed surface, so we cannot ...
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39 views

Exterior derivative of local basis element $dx^k$ is zero

Let $M$ be a smooth manifold and let $\omega = \sum_{(i_1, \dots, i_n)}f_{(i_1, \dots, i_p)} dx^{i_1} \wedge \dots \wedge dx^{i_p}$ be a differential $p$-form. Let $d$ denote the exterior derivative. ...
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2answers
35 views

Tangent space of a manifold is a vector field?

This is a follow up question on an answer to my previous question. Let $M$ be a smooth $n$ manifold and let $U\subseteq M$ be a domain. Let $T_xU$ denote the tangent space to $U$ at point $x$. Let ...
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Question regarding character varieties on a torus with compact gauge group

Let $G$ be a compact, connected Lie group. Let $x, y \in G$ be an arbitrary pair of commuting elements. Is there necessarily a torus $T \leq G$ containing $x$ and $y$? Apparently not: Commutativity ...
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Torsion vs Curvature of a curve

I was wondering if there is a simple explanation of the torsion and curvature in $\mathbb{R}^3$ of a curve. The curvature measure somehow the acceleration perpendicular to the tangent vector, but ...
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55 views

Two surfaces are not isometries of each other, but have the same Gaussian Curvature

How can you show that two surfaces are not isometries of each other, but have the same Gaussian Curvature. For example, I see that: the helicoid given by X = (ucosv, usinv, v) & the ...
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25 views

How to Compute the Torsion and Curvature of a Parametric Curve

So I have a parametric curve $\bf{r}=${$x(n),y(n),z(n)$} such that the functions $x(n)$, $y(n)$ and $z(n)$ are polynomials of $4$-th degree. I have several of these curves, and I want to calculate the ...
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30 views

Finding global sections of a vector bundle over a compact manifold which generate each fiber

The following is an exercise I am doing for review for a midterm exam in differential geometry. Question: Let $\pi:V\to X$ be any real differentiable vector bundle of rank $r$. Assume that $X$ is ...
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14 views

Bianchi identity number of independent equations

What is the number of independent equations of the second Bianchi identity: $$R_{abcd;e}+R_{abec;d}+R_{abde;c}=0$$
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36 views

Statistical Inference, Differential Geometry and Entropy

Context: Statistical Inference and Differential Geometry Let's consider a generic $ p(x;\theta) $ distribution with $ \theta $ Parameters Vector, it is obvious that $$ \int p(x; \theta) dx = 1 $$ ...
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Application of constant rank

Let $M^m$ and $N^n$ be differentiables manifolds, where $m$ is dimension of M and $n$ is dimension of $N$. If $f:M^n \to N^n$ is smooth map, with constant rank, show that: a)If $f$ is injective ...
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19 views

Find the area of parallel surface

Q: Consider a surface $M$ with regular parametrization $X:U_{open}\subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ and define the parallel surface $M_t$ by $$Y(u,v)=X(u,v) + tN(u,v)$$ where $N(u,v)$ ...
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If $\operatorname{Def} X = \frac{1}{2} \mathcal L_X g$ then what is $\operatorname{Def}^* \operatorname{Def} X$?

Let us define a deformation operator $\operatorname{Def}$ on a Riemannian manifold $(M,g)$, acting on divergence-free vector fields as $$ \operatorname{Def} X = \frac{1}{2} \mathcal L_X g, $$ where ...
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What is the definition of $dx$

I have just started to study differential forms. I don't yet fully understand the definition of what a differential form is (it's a $p$-times covariant tensor field) but I know that if $U$ is an open ...
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1answer
27 views

Basic diff.geometry question: Understanding coordinate charts by example

I recently learned the notion of coordinate chart: If $M$ is a manifold and $U\subseteq M$ is an open set in $M$ then a coordinate chart would be a smooth homeomorphism $\varphi : U \to V \subseteq ...
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Can a region always be parametrized by a single function?

Can some connected region in $\Bbb R^n$, possibly with some other nice conditions on the region, always be parametrized by a single function $\vec r(x_1, x_2, \dots, x_k)$ (even if it may be easier to ...
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A question related to the topology of the level sets of a particular type of smooth functions $f:\mathbb{R}^2\to \mathbb{R}$.

Let $f:\mathbb{R}^2\to\mathbb{R}$ be a smooth function without critical points; i.e. such that $\nabla f(x)\neq (0,0)$, for all $x\in\mathbb{R}^2$. Is it true or false that all the level curves of $f$ ...
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Basic question: Condition for a map associated to a linear series to be an immersion

I am reading this set of lectures of a class by Prof. Harris. There is a theorem. Let $X$ be a Riemann surface and $\phi:X\rightarrow\mathbb{P^r}$ be the map defined by a linear series without ...
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1answer
44 views

Riemann curvature product metric

Suppose that $M=M_1 \times M_2,$ with the product metric $g= g_1 \oplus g_2.$ Let $p\in M$ and suppose that $X \in T_pM_1$ and $Y\in T_pM_2.$ I want to show that $R(X,Y,Y,X)=0,$ at the point $p.$ I ...
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Calculation in From Seiberg Witten to pseudo-holomorphic curve

I am reading the Taubes's paper: From From Seiberg Witten to pseudo-holomorphic curve. I don't know how to get the result (2.17) \begin{eqnarray*} ...
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18 views

Pullback of Euclidean metric under spherical coordinates

I came across the following problem when I was reading a paper. Let $\Omega=B_2\subset\mathbb{R}^3$, which is the ball of radius 2 centered at origin. Map $F:B_2\backslash \{0\}\to B_2\backslash B_1$ ...
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2answers
49 views

Topologies in a Riemannian Manifold

I'm studying Differential Manifolds using Manfredo do Carmo's Book (Riemannian Geometry) and although I see no mention of this in Do Carmo's book, it's really easy to see a Riemannian Manifold as a ...
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Compare three tangent vectors constructed via parallel transport, exponential map and Jacobi vector field respectively

Given a Riemannian manifold $X$, a point $x\in X$ and $u,v\in T_xX$, I wanted to compare the following three vectors in $T_{\exp_x(v)}X$. $u_1=$ The parallel transport of $u$ along the geodesic ...
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1answer
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Quaternions $\Leftrightarrow $ Rotations -Conceptual question

I have an angular velocity vector analogue defined as(called Darbaoux vector some times in curve) $ \omega(s)=\frac{\mathrm{d}\theta(s) }{\mathrm{d} s}=\delta\theta(s)_x \vec{i}+\delta\theta(s)_y ...
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1answer
28 views

Getting Ricci Curvature From $g_{ab,cd}$

How does one see that $$c g_{ab,cd}\left(\eta^{ac}\eta^{bd} - \eta^{ab}\eta^{cd}\right)$$ is equal to $$(c/2)\eta^{bc}\eta^{ae}\partial_{a}\left(g_{be,c} + g_{ce,b} - g_{bc,e}\right) - ...
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Show that in these coordinates M is locally the graph $z=f(x,y) = \frac 12(k_1x^2 + k_2y^2) + e(x,y)$

Let us say that P is the origin and TpM is the tangent plane that is the xy-plane. We will let the x,y axes be the principal directions at P. Also, we will let the limit $$\lim_{(x,y)\to ...
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on special Kähler manifolds

Take Lie group $G$ with some hypotheses (compact, connected, maybe semisimple); call its Cartan subalgebra $\mathbf t \subset \mathbf g$ and Weyl group $W$. Why does the construction $\mathcal ...
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32 views

Gaussian curvature of one sheet hyperboloid

Q: Consider an one sheet hyperboloid $S$ sitting in $\mathbb{R}^3$ which defined by $x^2+y^2-z^2 =1$. Show that there is a straight line in $S$ through every point of $S$. Also, deduce without any ...
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Are the orbits of a connected Lie group acting on a vector space always embedded manifolds?

Setting: We have a connected Lie group $G$ and a smooth map $G \to GL(V)$, where $V$ is a finite-dimensional vector space. Are the orbits of $G$ on $V$ embedded submanifolds? More precisely, if one ...
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1answer
24 views

Deriving of the Jacobi bracket and the chain rule

This is from a passage that derives the Jacobi bracket from first principles. I cannot understand how the first equality works. It seems to use the chain rule and I agree with the second term but ...