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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This equation define a regular surface?

Consider the function: $f(x,y,z)=xyz^2$ Its gradient is $\nabla f=(yz^2, xz^2, 2xyz)$ then the critical values are all in the sets $\{(x,y,0): x,y\in \mathbb{R}\}, \{(0,0,z): z\in \mathbb{R}\}$. My ...
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19 views

Cheeger constant for $S^2$

I want to calculate explicitly Cheeger constant for $S^2$, but I haven't found any sources or examples. I'm using this definition $$h(M)=\inf_A\{\frac{vol_{n-1}(\partial A)}{vol_n{(A)}}:vol_n(A)\leq ...
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Lie Bracket, Hopf Fibration, independence of choice

Let $S^3$ be the standard sphere (with the metric induced from $\mathbb{R}^4$) and $\pi\colon S^3\rightarrow \mathbb{C}P^1$ the hopf fibration. Equip $\mathbb{C}P^1$ with the Fubini-Study metric so ...
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34 views

The isomorphisms between $S^5$ and $SU(3)/SU(2)$?

What is the precise isomorphisms between the coset $SU(3)/SU(2)$ and the five-sphere $S^5$?
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1answer
16 views

Subspace not open of a differentiable manifold

Suppose $M$ is an orientable differentiable manifold with dimension $n$. $U$ is a subspace of $M$. If $U$ is not open, is it true that $U$ also is an orientable differentiable manifold ? I need a ...
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38 views

Simple Vector Space Rotation

Given Data in the problem & notation convenstions We have 3 rotation vectors called $\vec{\theta_1},\vec{\theta_2},\vec{\theta_3}$, magnitude of these vectors will give you angle of rotation We ...
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22 views

boundary of a 3-cell

Let $I^k=[0,1]^k$. I want to calculate $\partial (I^3)$ rigorously. In case of $I^2$, one can easily separate as $\partial I^2 =\partial\sigma_1+\partial\sigma_2$, where $\sigma_1=[0,e_1,e_2]$ and ...
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25 views

Loxodrome : found an error on wolfram MathWorld web site?

Could it be?... We find this claim on Wolfram MathWorld site http://mathworld.wolfram.com/SphericalSpiral.html The claim is that this curve (given in oblate spheroidal coordinates in the limit where ...
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35 views

What is a moduli space for a differential geometer?

A moduli space is a set that parametrizes objects with a fixed property and that is endowed with a particular structure. This should be an intuitive and general definition of what a moduli space is. ...
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53 views

About the term $-\nabla_{[u,v]}w$ in the definition of Riemann curvature tensor

As we know, in the definition of Riemann curvature tensor, we require $$ R(u,v)w=\nabla_u\nabla_v w-\nabla_v\nabla_u w-\nabla_{[u,v]}w $$ Could somebody tell me why we need $-\nabla_{[u,v]}w$ ...
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1answer
46 views

Fixed point set defined by an isometry is a geodesic

The question is asking to me prove that: Consider a fixed point set $F=\{x \in S : f(x)=x\}$ in a smooth Riemannian surface with $f:S \rightarrow S$ be an isometry. If $F$ is a smooth 1-manifold, then ...
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22 views

Restriction of Poisson bracket on symplectic leaf

Consier $\mathbb{R}^3$ endowed with the following Poisson bracket $$\{x_1, x_2\}=x_3, \,\,\, \{x_2,x_3\}=x_1, \,\,\, \{x_3,x_1\}=x_2.$$ Let $Z=x_1^2+x_2^2+x_3^2=r^2$ be a symplectic leaf of this ...
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1answer
63 views

Parametrising the unit circle without sine and cosine

Is there a nice way to make a smooth and periodic parametrisation $\gamma\colon\mathbb R\to S^1$ of the unit circle $S^1$ in $\mathbb R^2$ that does not somehow involve sine/cosine or (what I find to ...
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38 views

Is there a surface S$\subset R^3$ whose Gaussian curvature is -1 at each point S?

Is there a surface $S\subset \Bbb R^3$ whose Gaussian curvature is $-1$ at each point $S$? At first I think this does not make a sense. But googling and googling.. I found a 'final exam problem' ...
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27 views

Any hint to show this flow property?

I just wonder if someone might help with this exercise. Let $X$ be a vector field on a compact smooth manifold $M$ and let $\phi_t$ be the flow of $X$. Show that for all $x \in M$, then ...
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Differential Geometry [on hold]

I have this old midterm as an assignment and am a bit lost. Any help on any of these would be very much appreciated! I'm new to this community and am excited to be apart of this site. Here is the ...
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34 views

Need a reference book on stokes theorem other than rudin

As the title suggests, I am looking for a book (other than Rudin's Mathematical Analysis) that covers differential forms, simplexes, chains, stokes theorem. Actually I am not familiar with tensor ...
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30 views

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 [on hold]

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
64 views

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
23 views

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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63 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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44 views

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
15 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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50 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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2answers
31 views

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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1answer
19 views

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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19 views

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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11 views

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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24 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
40 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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42 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
56 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
31 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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1answer
48 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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21 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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1answer
51 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
37 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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25 views

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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1answer
39 views

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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64 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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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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32 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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15 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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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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17 views

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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1answer
27 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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26 views

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 ...