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Questions tagged [differential-geometry]

Differential geometry is the application of differential calculus in the setting of smooth manifolds (curves, surfaces and higher dimensional examples). Modern differential geometry focuses on "geometric structures" on such manifolds, such as bundles and connections; for questions not concerning such structures, use (differential-topology) instead. Use (symplectic-geometry), (riemannian-geometry), (complex-geometry), or (lie-groups) when more appropriate.

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How do I generalize the dot product (bilinear form) in spherical coordinates?

In cartesian coordinates, the unit vectors $\{u_x, u_y, u_z\}$ are universal. That is, $u_x(x, y, z)$ is constant and so on for the rest of them. Because of that, the dot product $\langle v | w \...
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12 views

Zeros of a vector field on certain complex manifold. [on hold]

Suppose that $X$ is a 2 dimensional complex manifold $X$ contains no curve There is a non trivial holomorphic vector field $X$ contains no 1 dimensional complex sub manifolds. The Euler number of $X$...
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19 views

Confused about induced connection definitions, pullback bundle

After reading a lot on induced connections, I am quite confused - I must admit I am a beginner lacking formal mathematics background. Essentially there appear to be two definitions which I am trying ...
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0answers
11 views

Can embeddings be categorized into pre-embeddings and post-embeddings?

Consider a spin $1/2$ particle that moves on a two-dimensional spherical surface, and examine its orbital angular momentum in quantum mechanics. Approach 1: Take the surface as a Riemann surface, ...
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10 views

Smooth regularity of Lindelof smooth manifolds modeled on smoothly regular convenient vector spaces

The following statement is the first statement of the corollary in Section 27.4 of A. Kriegl, P.W. Michor, The convenient setting of global analysis: If a (smoothly Hausdorff) smooth manifold is ...
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14 views

Generic maps and double points

What is the definition of a generic map between two smooth manifolds? Are all continuous maps between smooth manifolds homotopic to generic ones? Let $N^n, M^{2n}$ be compact smooth manifolds. Why ...
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0answers
19 views

Vanishing of rotation of timelike unit vector in numerical relativity

I'm a bit confused. I don't think it is that difficult, but still don't manage :-( So the question is why the rotation of the timelike unit-vector in numerical relativity $$n_\mu=-\alpha \nabla_\mu t$...
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2answers
161 views

How to compute a Jacobian using polar coordinates?

Consider the transformation $F$ of $\mathbb R^2\setminus\{(0,0)\}$ onto itself defined as $$ F(x, y):=\left( \frac{x}{x^2+y^2}, \frac{y}{x^2+y^2}\right).$$ Its Jacobian matrix is $$\tag{1} \begin{...
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0answers
19 views

General method for parametrising as a ruled surface or showing a surface to be ruled

I'm sorry for asking such a basic question, but a practice question I'm currently completing asks for me to give a local parametrisation of $H = \{(x,y,z) \in \mathbb{R^3} \vert x^2 - y^2 + z^2 = 1\}$ ...
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0answers
14 views

covariant divergence of mixed tensor

Does anyone know how to solve this problem? Show that for a symmetric tensor $T^{\mu\nu}$, the covariant divergence of the mixed tensor ${T_{\mu}}^{\nu}$ is given by a somewhat more complicated ...
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2answers
47 views

Product of vector fields is not a vector field

Let $M$ be a manifold and $X,Y$ be vector fields on $M$. The bracket $[X,Y]:=XY-YX$ is a vector field when $X,Y$ are smooth, but why is $XY$ not a vector field when $X,Y$ are smooth? By definition, a ...
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0answers
18 views

Notions of Convexity

Let $(S,g)$ be a complete Riemannian manifold and $M\subset S$ an embedded submanifold (of the same dimension) with non-empty boundary $\partial M$. I am interested in understanding the relation ...
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1answer
24 views

Limit of Laplacian of the distance at the origin

Let $p$ be a point in a Riemannian manifold $M$ and $d_p$ be the distance from the point $p$. Prove that $\lim_{x\rightarrow p}\Delta d_p(x)=\infty$ I can easily prove it in $\mathbb{R}^n$. But for a ...
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0answers
15 views

fields and cones

I have a euclidean cone, $X$, with some cone point $P$. I assume that a vector field on $X - P$ is parallel if an isometry that carries an open set $X - P$ to $\mathbb R^2$ carries the vector field ...
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0answers
23 views

$M^n\subset \mathbb{R}^{n+1}$ isometric immersion, $R\equiv 0\Rightarrow M$ locally isometric to $\mathbb{R}^n$?

Let $f:M^n\to\mathbb{R}^{n+1}$ be an isometric immersion with $R\equiv 0$, where $R$ is the curvature tensor of $M$. Can we say that $M$ is locally isometric to $\mathbb{R}^n$? I know that by ...
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0answers
16 views

Restriction of vector field to the circle

I'm working on a problem that asks me to describe a differentiable manifold structure for the circle $S^1$ and then to calculate the restriction of the vector field $$\xi=-y\frac{\partial}{\partial x }...
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0answers
24 views

Derivative of a pullback and commutator

I'm having difficulties trying to proof this derivative at t= 0: $$\frac{d}{dt} \phi_t^{*} (gY) = [X, gY] $$ where $g$ is a function, $\phi^{*}_t$ is the pull-back by the small parameter $t$ and $X,...
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0answers
23 views

Is the curvature the exterior covariant derivative of the connection?

Let $P\to M$ be a $G$-principal bundle, $G$ a topological group, $\omega$ the connection and $V$ a vector space. We define $d_\omega: \Omega^k_G(P, V)\to\Omega^{k+1}_G(P, V)$ the $\textit{exterior ...
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0answers
27 views

Questions on coordinate representation of a map between smooth manifolds (sphere with stereographic projections)

On the sphere $S^n$, consider the atlas given by the stereographic projection: $\{(U_1, \phi_1), (U_2, \phi_2)\}$, where $U_1 = S^n \backslash \{N\}$, $U_2 = S^n \backslash \{S\}$, $N$ and $S$ are ...
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18 views

Weil Model of Equivariant Cohomology

I was reading this paper, and was stuck on a supposedly trivial calculation at page 13. I have trouble understanding the authors' calculation of $d_X$. The authors claimed $D\lambda a= D(a-i(X)a\...
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1answer
37 views

Manifolds that admit Lorentzian metrics?

John Lee says in "Riemannian Manifolds: An Introduction to Curvature": With some more sophisticated tools from algebraic topology, it can be shown that every noncompact connected smooth manifold ...
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0answers
12 views

Bounding the symbol of a differential operator

Consider a bundle map $P$, between the bundles $\Bbb R^m \times \Bbb R^n \rightarrow \Bbb R^m$ and $\Bbb R^m \times \Bbb R^k \rightarrow \Bbb R^m$. On each fiber above $\xi \in \Bbb R^m$, $P$ acts as ...
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0answers
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Degree of Line bundle and Intersection

Let $(\Sigma,g)$ be a closed Riemann surface with metric $g$. For any holomorphic line bundle $L\to \Sigma$, given a metric we have its curtature in terms of Chern connection $A$. It is well-known ...
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1answer
24 views

Is $\{(x,y,z)\in\mathbb{R}^3:x^3+y^3-z^3=1 \}$ a regular surface?

I think the answer is yes by Proposition 2.2 of Do Carmo, which states the pre-image of regular values of functions of the form $f:\mathbb{R}^3\rightarrow \mathbb{R}$ is a regular surface. $$S=\{(x,y,...
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Relationship between locally trivial and locally non vanishing section.

What I want to show is that If a line bundle (not necessarily locally trivial) has locally nonvanishing sections for each open set in the open cover of the base space, then it is locally trivial. ...
4
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1answer
38 views

Cartan homotopy formula and curl

In Topological Methods in Hydrodynamics, V. I. Arnol'd writes that the following expression $$curl(\mathbf a \times \mathbf b)=[\mathbf a, \mathbf b]+ \mathbf a \ div \ \mathbf b - \mathbf b \ div \ \...
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0answers
82 views

Are two circles of different diameters similar on a sphere? [on hold]

The teacher told me that there are only congruent triangles on the sphere, but no similar triangles. My question is, are the two circles with different diameters on the sphere similar? I'm asking a ...
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0answers
15 views

Difficulty understanding how to construct different smooth structures on a manifold using the identity map

I am reading a text stating: “It is easy to give examples to show that a $C^{\infty}$ homeomorphism need not be a diffeomorphism. For any integer $n > 1$ the map $x^{2n-1}: \mathbb R \to \mathbb R$...
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0answers
31 views

Computation of the push forward of vectors

I am trying to understand the push forward of a vector field by going through a specific calculation. Consider $f: \mathbb{R}^3 \rightarrow \mathbb{R}^2$ given by $f(x,y,z) = (x+y+7, z-x-5)$ and ...
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1answer
18 views

Relation of Killing's vector field, Spacetime dimension, and Riemannian curvature tensor

I am a undergraduate student who studying general relativity. While I am reading numerical relativity written by Masaru Shibata, I do not understand how to derive the equation $\nabla^{a} \nabla_{a} \...
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1answer
32 views

Differentiable structure in $S^1$ such that the inclusion map is differentiable

I am asked to explicitly describe a differentiable structure on $S^1$ (with the usual subspace topology) such that the inclusion map $\iota: S^1 \to \mathbb{R}^2$ is differentiable. I've seen some ...
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0answers
17 views

Looking for a class name and a potential general solution

Is there a standard class name if any of the following differential equation there similar to Laplace equation and equivalent solution based on legendre polynomial, or spherical harmonics? $\partial^...
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0answers
17 views

Derivation of the Geodesic Equation from Hamilton-Jacobi Theory

I'm trying to prove a result in my general relativity class and I'm confused. If we have the Hamiltonian: $$ H = g^{\mu\nu}p_\mu p_\nu + m^2 $$ Subject to Hamilton's Equations: $$\frac{\partial x^\mu}{...
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1answer
31 views

Covering map of the universal cover $\widetilde{G} \rightarrow G$ for $G$ a Lie group is a homomorphism?

In a paper I'm reading, we have a compact Lie group $G$ and he says "We can identify $\pi_1(G)$ with the kernel of $\widetilde{G} \rightarrow G$. I can't seem to find anything that says that the ...
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0answers
15 views

How to calculate the geodesic curvature of a discrete 3D curve?

I have coordinates of a set of points that form a closed loop that lies in a 3D surface. I know the equation of the surface and I can calculate it's surface normal at any point. I found that for a ...
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1answer
28 views

Almost complex structure question

I know that if $M$ admits an almost complex structure $J$, then $\text{dim}_{\mathbb{R}}(M)=2k$, thus every odd-dimensional manifold doesn't admit an almost complex structure. My question is, are ...
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1answer
24 views

Tangent line of non-simple curves

I have a small confusion about curves. Given a non-simple parametric curve $\alpha : I \mapsto \mathbb{R}^{2}$. How does the tangent vector $T(s)$ and the tangent line $\{\alpha(s) + t T(s) : t \in \...
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0answers
23 views

Curvature forms as exterior covariant derivative?

I have read on several forums like this one, that given a connection form $\omega$ on a principal bundle and its curvature form $\Omega$, I can state that $\Omega=d_\omega\omega$ alike I do in the ...
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0answers
13 views

Partial Derivatives - Covectors - Antisymmetric Tensors

Suppose that $V_{\mu} (x)$ transforms like a covariant vector under a general curvilinear coordinate transformation $x'^{\nu} = x'^{\nu} (x^{\mu})$ . Show that the partial derivatives $V_{\mu,\nu} \...
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1answer
37 views

if a tangent at a point of a manifold taken as a function is zero at two points, it is zero at all points

Let $M$ be a manifold and $m\in M$. We call $t$ a tangent at $m$ if for every pair $(f,g)$ of smooth real functions defined in a neighborhood of $m$ and for every pair $(a,b)$ of real numbers, we have ...
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1answer
28 views

How to find an embedded submanifold in $\mathbb{R}^{n}$ which pass through some given points in $\mathbb{R}^{n}$ and has the lowest dimension?

How to find an embedded submanifold in $\mathbb{R}^{n}$ which pass through some given points in $\mathbb{R}^{n}$ and has the lowest dimension? I have no idea with this question. I know, at least, the ...
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1answer
17 views

Question on Solid Angles and Linking Number

In Spivak's Calculus on Manifolds there is a question about solid angles and linking number that confuses me. Suppose $f([0,1])=\partial M$ where $f$ is a closed curve and $M$ is a compact oriented ...
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0answers
21 views

Computing all possible conformal factors on the sphere

Proposition to prove. Let $\tau\colon \mathbb S^n\to \mathbb S^n$ be a conformal map, meaning that $\tau^\star g= \Lambda^2 g$ for a scalar field $\Lambda$. (Here $g$ denotes the standard metric ...
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1answer
37 views

Integration on Differential Forms.

So I know that divergence and curl of a vector field $F$ can be related to a differential form $\alpha$ by div $ F = \star d \star \alpha$ and curl $ \cdot F = \star d \alpha $ , where $\star$ is the ...
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0answers
8 views

Surface Area of generalized polar parametrization

Suppose we have a smooth domain $D$ in $\mathbb{R}^n$, which can be parametrized by polar coordinates $(\zeta, r(\zeta))$. Here, $\zeta$ denotes a point on the $(n-1)$ dimensional unit sphere $B$ and $...
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0answers
53 views

Existence of a countable locally finite cover with nonempty intersection of two adjacent elements

Let $\Omega$ be an open connected set in $\mathbb{C}$, not necessarily bounded. Does there exist a countable locally finite cover of $\Omega$ consisting of only open discs $\{ B(z_i, r_i): i\geq 1\}$ ...
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0answers
33 views

Phase, Isochrons, Isochrons map and Lift

at the moment i read the following paper: https://arxiv.org/pdf/1512.04436v1.pdf I have some questions about it and i hope someone can help me. On page 4/5 they introduce isochrons and the isochron ...
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1answer
59 views

Questions on Cartan's magic formula $\mathcal{L}_X=i_X \circ d + d\circ i_X$

Algebra $A$ is called graded algebra if it has a direct sum decomposition $A=\bigoplus_{k\in\Bbb Z} A^k$ s.t. product satisfies $(A^k)(A^l)\subseteq(A^{k+l}) \text{ for each } k, l.$ A ...
4
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2answers
30 views

Circle inscribed between two curves

Consider the plane region $S_n$ bounded from above and below for the graphs of $f_n(x)=x^{1/n}$ and $g_n(x)=x^n$, $0\le x\le1$. How to find the radious and center of the circle inscribed in $S_n$? ...
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1answer
28 views

Do Carmo Section 2.2 Proposition 2 Proof Clarification

I am having some difficulties in understanding the proof given by Do Carmo of the following proposition. I will run through the proof and comment on it as it progresses and place my questions ...