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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Strainers in Alexandrov spaces

I am reading the section on Strainers in Burago, Burago and Ivanov's book "A Course in Metric Geometry". I have been struggling with the proofs of some of the lemmas. On Lemma 10. 8. 13, the authors ...
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61 views

The property of an integral manifold of differential form

Let's define the $1$-form $\eta$ on $\mathbb{R}^3$ by formula: $$\eta = A(x,y,z)\;\mathrm{d}x + B(x,y,z)\;\mathrm{d}y + \mathrm{d}z \text{.}$$ And let's assume that $$\mathrm{d}\eta \wedge \eta = ...
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70 views

Proof behind $S^n\cong SO(n+1)/SO(n)$

I have been trying to understand the fact that $S^n \cong SO(n+1)/SO(n)$. I believe I have the intuition correct at this point; consider the case when $n=2$ as we have $S^2 \cong SO(3)/SO(2)$.: We ...
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1answer
24 views

Applying Green's theorem for a line integral of a vector field

Integrate the vector field $F(x,y)=(e^y+\frac{1}{y+3},xe^y-\frac{x+1}{(y+3)^2})$ over a curve that goes from $(-1,0)$ to $(-1,2)$ to $(0,1)$ to $(1,2)$ (in a linear fashion). Now, I'm almost certain ...
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26 views

Showing that the two-sheeted cone is is not a regular surface

I was given the following problem in class: Show that the two-sheeted cone, with its vertex at the origin, that is, the set $$\{(x, y, z) \in \mathbb{R^3}\,|\, x^2 + y^2 - z^2 = 0\}\text{,}$$ ...
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De Rahm Cohomology of Complex Grassmannian

Since the complex Grassmannian $G_k(\mathbb{C}^n)\cong SU(n)/S(U(k)\times U(n-k))$ is connected and simply connected, the first two de Rahm cohomology groups are given by $$ ...
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56 views

Accepted symbol (or way of writing) “A is a subset of B or B is a subset of A”

I am looking for a concise way to write the statement "$A$ is a subset of $B$ or $B$ is a subset of $A$". The context is the Grassmannian and two elements $A,B\in G_k(\mathbb R^n)$ in it. The two ...
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48 views

Visualizing the level sets for this function

Let $F: T^{2} \to \mathbb{R}$ be given by $(x_{1}, x_{2}, x_{3}) \mapsto x_{2}$. Recall that $$T^{2} = \{(x_{1}, x_{2}, x_{3}) : \left(\sqrt{x_{1}^{2} + x_{2}^{2}} - R \right)^{2} + x_{3}^{2} = ...
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23 views

Terminologies for induced connections

Given a Riemann manifold with a Kozul/Affine connection, if you take any subbundle of the tangent bundle there is an induced connection given by applying the ambient Kozul connection and projecting to ...
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114 views

Connected sum in an ambient space

Let $M$ be a smooth c connected $m$-manifold, $N_1$, $N_2$ two smooth disjoint connected $n$-submanifolds with boundary, both contained in the interior of $M$ (if necessary). Assume that $M\setminus ...
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35 views

Homology of smooth loop spaces of spheres

I'm looking for a reference on the homology of $C^\infty(S^1,S^n)$. So far I've only been able to find references for spaces of maps which are $C^k$ or in a Sobolev class. I also have references which ...
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33 views

Computing geodesic distances from structural data

I am attempting to compute geodesic distances on manifolds where structural data have been sparsely sampled. First, off I am not well versed in the mathematics of differential geometry but I do have ...
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1answer
60 views

Why $S^1\times S^{2m-1}$ carries a complex structure.

Let $S^n$ denotes $n$-sphere, then why $S^1\times S^{2m-1}$ carries a complex structure.
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30 views

Dimensions of the cohomology groups of certain complicated space

Let contruct the space $X$. We take the complex projective space $\mathbb{C}P^2$, pick two points $p_1, p_2 \in \mathbb{C}P^2$ and remove two small, disjoint, open $4$-balls $B_j$ centered at $p_j$. ...
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57 views

Principle G bundles v.s. Flat G connection

What is the difference between Principle G bundles v.s. Flat G connection? I heard that for a discrete group $G$ (in physics, or a finite group $G$ in math), the principle G bundles is the same ...
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88 views

Path to Differential Geometry

What do I need to learn to start on the rigorous study of differential geometry? I'm about to start my 3rd undergrad year at school, and have taken Cal 1-3, Linear Algebra, Elementary Number Theory, ...
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32 views

Stoke's theorem explanation

Firstly, Wikipedia informs me that Stoke's theorem might mean two different things so I'm pretty sure I'm talking about the general one. I'll just state what I understand from it and I'd like someone ...
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26 views

Number of Umbilical points

Studying umbilical points on a surface, I wonder if for a convex surface there will be some similar formula to the four vertex theorem, ie will exist some minimum number of umbilical points for $ S $ ...
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89 views

is the stress tensor (elasticity theory) actually a pseudo tensor?

Please, is the stress tensor (elasticity theory) actually a pseudo tensor? It seems to me it must change its sign when coordinate system changes its orientation. The argument is as follows: We must ...
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1answer
21 views

How compute the first fundamental form by using the matric tensor transformation

Consider the parametrization of the unit sphere: $\overline{X}=\left( \cos{a_1} \sin{a_2}, \sin{a_1} \sin{a_2}, \cos{a_2} \right)$ Given the parametrization $\overline{\Pi}:\mathbb{S}^{2} - N ...
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21 views

Differentiable structure on the gauge group?

In this paper I have come across a formulation involving differentiation in the gauge group of a principal bundle which I do not understand (found at the very top of p. 369). Let $P\rightarrow M$ be ...
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21 views

convex curve equivalent definition

I have seen two different version of definition of a closed convex curve in a plane: For a curve $r(t)=(x(t),y(t))$ 1.The whole curve lies on only one side of any tangent of the curve 2.Any ...
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17 views

Verification on Intersection of Tangent Lines.

Given a function $\gamma (t) = (-1,4)(2-3t)^2 + (1,0)(3t - 1)^2 $ figure out the control point $P1$ which exist at the intersection of the tangent lines of $P0$ = $\gamma (0) $ and $P2$ = $\gamma (1) ...
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29 views

Find the areas $A(D)$ of $D$ and $A(\sigma(D))$ of $\sigma(D)$.

A regular parametrized surface $\sigma:U\to\mathbb{R}^{3}$ is given where $U=\lbrace (x,y)\in \mathbb{R}^{2}\mid (x,y)\neq (0,0)\rbrace$. Only the coefficients of the first fundamental form are known ...
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46 views

Integral notation from cartesian from polar coordinates

Given an integral $$I=\int\limits_{\mathbb{R}^n} \cdot \; dx,$$ we can introduce polar coordinates, such that $$I=\int\limits_{\Bbb S^{n-1}} \cdot \; d\theta.$$ Another way to express the latter one ...
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17 views

Citing a result on obstruction to Lagrangian Embedding

Let $L$ be a closed orientable Lagrangian embedding in $\mathbb C^n$. Then $\chi(L) = 0$, where $\chi(L)$ denotes the Euler characteristic of $L$. This fact is more or less stated in section 3.2 of ...
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49 views

Decomposition of Laplacian into tangental and normal components w.r.t. submanifold

If I have the covariant Laplacian operator acting on a tensor e.g. $\nabla^2 h_{\mu\nu}$ on a (pseudo-Riemannian) manifold, and I have (say a codimension-2) submanifold, how can I "decompose" the ...
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131 views

Is every scalar differential operator on $(M,g)$ that commutes with isometries a polynomial of the Laplacian?

On $(\mathbb{R}^n, g_{\text{std}})$ with $\Delta$ the Laplacian, the following holds: Fact: Every scalar differential operator $D$ that satisfies $D \circ F^* = F^* \circ D$ for all isometries $F ...
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55 views

Wedge product and determinants

I am attempting to self-study differential forms this summer, but I ran into this definition for the wedge product in my book, and it doesn't make any sense to me. Note that $\Bbb R^3_p$ is the ...
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74 views

Covariant derivative in cylindrical coordinates

I am confused by the Wolfram article on cylindrical coordinates. Specifically, I do not understand how they go from equation (48) to equations (49)-(57). Equation (48) shows that the covariant ...
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3answers
64 views

A diffeomorphism with negative Jacobian swaps the orientation?

Let C be a simple close oriented curve $C^1$ in $\mathbb{R}^2$ and let $g: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ a diffeomorphism such that $\forall (x, y) \in C$ it holds that the determinant of the ...
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51 views

Establishing compactness of manifolds for the purposes of applying Chern-Gauss-Bonnet

A unit sphere possesses an induced metric, $$ds^2=d\theta^2 + r^2\sin^2\theta d\phi^2$$ By applying the Cartan formalism, for a basis $e^\theta = d\theta$ and $e^\phi=r\sin\theta d\phi$, I found, ...
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50 views

Global and local coordinates on a manifold, and their relations to curvature

I would be pleased to have some information about coordinates in differential geometry. A) First I would like to check whether or not the definitions I use are correct. (Mainly for the sake of ...
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1answer
34 views

Natural connection on tautological bundle over real Grassmannian

Let me get to the point immediately: Is there a natural connection on the tautological vector bundle over a Grassmannian (of a real vector space equipped with an inner product)? In a paper I'm ...
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1answer
64 views

About reparametrization of timelike curves in $\mathbb{L}^3$ (Lorentz-Minkowski space)

I think there is something wrong with the proof this text gives of Lemma $2.1.5$, in pages $19$ and $20$, for timelike curves. I used another function, and it seems to work. Either I'm wrong, or he ...
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Explicit computation of spectrum of Hodge-Laplacian on forms

While I know of some explicit examples (sphere, flat torus) for the spectrum of the Hodge-Laplacian on 0-forms (i.e. the Laplace-Beltrami operator on functions), I haven't found anything for "actual ...
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27 views

arbitrary reparametrization

Let $\alpha: (a,b)\rightarrow \mathbb{R}^n$ of class $C^{\infty}$ with $\Vert\alpha^{\prime}\Vert>0 $ then if $\{ k,m,n\} \subset \mathbb{R}_+$ there repametrizacion $\beta: (m,n)\rightarrow ...
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1answer
72 views

Application of Riemann Roch

I have read that thanks to Riemann Roch theorem, if get $\Sigma$ a compact Riemann Surface of genus $g$ there exists a conformal branch covering $\phi: \Sigma \rightarrow S^2$ of degree less than ...
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1answer
39 views

Difference between flat of a vector and dual of a vector

The flat of a vector $X\in T_p(M)$ is defined as a dual vector $X^\flat\in T_p^*(M)$ given by the following map on vectors: $$ Y\stackrel{X^\flat}{\mapsto} g(X,Y) $$ The dual of a basis vector $e_j$ ...
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3answers
112 views

Eigenfunctions of the Laplace-Beltrami operator of a torus

The eigenfunctions of the Laplace-Beltrami operator of the flat torus $\mathbb{T}^2=\mathbb{R}^2/\mathbb{Z}^2$ and their multiplicity are well-known. What happens if we change the sides of the torus ...
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2answers
70 views

What is the calculus-theoretic formula to calculate the homotopy class/degree of a map $T^2\to S^2$?

I know by Hopf classification theorem that $[T^2;S^2]$(torus to sphere) are classified by the integral cohomology group $H^2(T^2;\mathbb{Z})\approx\mathbb{Z}$. Also I understand that in general, given ...
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Why do differential forms have a much richer structure than vector fields?

I apologize in advance because this question might be a bit philosophical, but I do think it is probably a genuine question with non-vacuous content. We know as a fact that differential forms have a ...
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26 views

Change of variables in calculating the integral of multivariable differential entropy

I know that for one dimensional differential entropy of a density function $p(x)$, one has the following formula by change of variables: $$H(p)=\int ...
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1answer
43 views

Why is the parallel transport on the 2-sphere SO(3)-equivariant?

I am trying to prove the following equation for $R \in SO(3)$: \begin{equation*} R^{-1}P_{R(\gamma)}(R(v)) = P_{\gamma}(v) \end{equation*} where $\gamma \colon \lbrack 0,1 \rbrack \longrightarrow S^2$ ...
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36 views

Computing integrals of differential forms

How do I compute the integral of the differntial form $\omega = xdy - y dx$ on $\mathbb R^2\backslash \{0\}$ along the path $\gamma(t) = (\cos(2\pi t),\sin(2\pi t))$? That is, what is ...
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107 views

existence of hemisphere

If $ \beta: (a,b)\rightarrow \mathbb{S}^2 $ is a simple closed curve such that $ \int_a^b\Vert\beta^{\prime}(t) \Vert dt<2\pi$ then there is an open hemisphere (or any rotation of this) containing ...
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36 views

Riemann Sphere: Holomorphic Functional Calculus

Why do we consider the holomorphic functional calculus on the Riemann sphere rather than the complex plane only? Is there a serious problem? Moreover isn't any curve encircling the spectrum ones ...
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73 views

universal covering of punctured plane and Poincaré metric

I want to prove the following result: Let $\Omega$ be the domain $\mathbb{C}\backslash \{a_1,a_2,a_3,a_4\}$. Its universal covering is the unit disk, and the standard Poincaré metric is pulled back to ...
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38 views

lower bound of a sectional curve

We say that a curve $ \alpha: I\rightarrow \mathbb{S}^2 $ is "sectional" if $ \alpha $ is a simple closed such that $\alpha$ divides $\mathbb{S}^2$ into two regions of equal area. I'm interested in ...
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17 views

Equivalence between pullback connections of smoothly homotopic maps

Let $f,g:M\rightarrow N$ be smooth maps between smooth manifolds such that there exist a smooth homotopy $H:M\times [0,1]\rightarrow N$ between them. If we have a principal bundle $P\rightarrow N$, we ...