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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Hodge star of second-rank antisymmetric tensor

Say we have a tensor $F$ which just for familiarity's sake, we take to be a second rank antisymmetric tensor. I understand that given the Hodge star operator defined as ...
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109 views

On the variation of a Kähler metric on a surface by pullback of the complex structure

Let $\Sigma$ be a compact, connected, oriented surface, and let $\rho\in\Omega^2(\Sigma)$ be a fixed volume form. Then any (almost) complex structure $J\in\Omega^0(M;\operatorname{End}TM)$ compatible ...
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2answers
85 views

Relationship between divergence operators defined with respect to two different volume forms.

Let us assume that you have a volume form $\mu$ defined on a manifold $\mathcal{M}$. Then you can define the divergence operator with respect to this metric, such that the following relationship holds ...
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35 views

A question about Lie derivative and taking trace

Let $(M,\omega)$ be a complex Kahler manifold, and $g$ is a smooth function such that $\int_Mg\omega^n=0$. It is obvious that there exists a smooth function $f$ such that $\triangle_\omega f=g$. ...
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1answer
23 views

Lie Derivative of Connection 1 form

On Page 106 of Kobayashi & Nomizu's 'Foundations of Differential Geometry', the authors write \begin{align*} (L_X \omega)(Y)&=X(\omega(Y))-\omega([X,Y]). \end{align*} Here, $\omega$ is the ...
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28 views

How to Induce a Metric on Homogeneous Space $G/H$ by the Metric from Bundle G

I am having a question on how to induce a metric $g$ on homogeneous space $G/H$, if one is given a ${\rm Ad}_H$-invariant metric $\bar{g}$ on G. More specifically and simply, consider principal ...
2
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1answer
68 views

Reference Request concerning Jet Bundles..

can anyone recommend me a nice reference concerning jet bundles? I've been looking for one for a long time but I couldn't find it...Thanks..
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1answer
33 views

Proof of Hopf's theorem using Liouville

Hopf Theorem A topological sphere immersed as a constant mean curvature surface in $\mathbb{E}^3$ is a round sphere In Heinz Hopf's Differential Geometry in the Large, a proof is given of the ...
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1answer
118 views

What is the origin of the terms 'jet' and 'prolongation' in differential geometry?

I am just curious what is the reason for the terms 'jet' and 'prolongation' in differential geometry? Is there some mental imagery that these names are supposed to evoke? Or are they so-named because ...
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1answer
102 views

Why is the momentum a covector?

Can someone tell me why the momentum is an element of the cotangent space? More detailed: if we have some smooth manifold M and the cotangent space $T_{x}M^{*}$ I know that the momentum p is an ...
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29 views

For which values of $c$ are the following sets become smooth manifolds

For which values of $c$ are the following sets become smooth manifolds 1) $\{(x,y)\in\mathbb{R^2} \mid x^3+xy+y^3=c \}$ 2) $\{(x_1,x_2,x_3,x_4)\in\mathbb{R^4} \mid x_1^2+x_2^2-x_3^2-x_4^2=0, ...
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1answer
64 views

Ricci tensor and average of a tensor

Let $(M^n,g)$ be an oriented Riemannian $n$- manifold and $g$ is a Riemannian metric on $M$ , $\mathrm{d}\sigma$ is Riemannian volume form on $S^{n-1}$ and $\text{Vol}(S^{n-1})$ is volume of ...
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1answer
46 views

Normal coordinate parallel along radial geodesics?

A radial geodesic in normal coordinates is given by $\gamma:t \mapsto t(V_1,....,V_n).$ Is it then true that any normal coordinate $\partial_x|_{\gamma}$ is parallel along $\gamma,$ i.e. ...
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2answers
89 views

How do the components of a cross product transform?

Let $x^{j}$ and $y^{k}$ be the components of two vectors $x,y\in \mathbb{R}^{3}$. According to the way the compontents of $x$ and $y$ transform when we change the basis, we know they are ...
2
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1answer
38 views

If $\phi: M_1 \to M_2$ a diffeomorphism between diff. manifolds, prove that if $M_2$ is oriented then so is $M_1$

Let $\phi: M_1 \to M_2$ a local diffeomorphism between two differentiable manifolds $M_1,M_2$. I want to prove that if $M_2$ is orientable so is $M_1$. Attempt: In order a manifold to be orientable ...
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1answer
55 views

Exercise about Christoffel symbols: prove that $\Gamma^k_{ij}(p)+\Gamma^k_{ji}(p)=0$

Let $\nabla$ be a linear connection on a manifold $M$. I want to prove that: $$\Gamma^k_{ij}(p)+\Gamma^k_{ji}(p)=0,$$ if in $p\in M$ normal local coordinates are defined. I have a solution which ...
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1answer
31 views

Continuation of a vector field on projective space

In the book of V.I. Arnold "Ordinary differential equations" there is a lemma which states that any vector field $F$ on $\mathbb{R}^n$ can be uniquely continued to a smooth vector field $\overline{F}$ ...
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11 views

Let be $K$ a tensor field, what is the relation between the coefficients of $K\big|_U$ and $\nabla K\big|_U$?

Let be $\nabla$ a linear connection on a manifold $M$ and $K$ a tensor field of type $(h,k)$. Using a local coordinate system $(U, \phi)$ we can write: $$ K\big|_U= \sum ...
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1answer
61 views

Sphere curvature as calculated from Liouville's equation

Liouville's equation for Gauss curvature tells us, that when Riemannian metric has the form $f^2(du^2+dv^2)$, then its Gauss curvature $K$ is expressed by the following equation: ...
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1answer
31 views

Need some help understanding the condition of the implicit function theorem

The condition for the implicit function theorem is that the (smooth) map $f: \mathbb R^n \to \mathbb R^m$ is locally a (smooth) map of $n-k$ variables if there are locally smooth maps $g_i , i \in ...
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1answer
48 views

About separation property of hypersurface

Let N be a complete Riemannian manifold and M be a complete hypersurface in N. M is said to have separation property if N\M is disjoint union of 2 connected open sets in N. Under what reasonable ...
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1answer
72 views

Groups that are not Lie Groups

What are some examples of groups that can not be given a smooth structure such that they become a Lie Group? Edit: To be a bit more specific, I was hoping that somebody could give an example of a ...
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1answer
52 views

Diffeomorphism $\phi : M \to M$. Why can it be written like this?

Let $\phi : M \to M$ be a diffeomorphism from $M$ to $M$. Let $v \in T_pM$ and $f$ a differentiable function near $\phi(p)$. Then we have $$ \Big( d\phi (v) (f) \Big)\phi(p) = v(f \circ \phi)(p) $$ ...
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1answer
28 views

Tangent and normal spaces of submanifold of fixed-rank matrices

Let $m \geq 2$. The subset $X$ of $m \times 2$ matrices with rank $1$ is a (smooth) submanifold of $\mathbb{R}^{m\times 2}$. Let $A$ be in $X$. I know from a more general statement that the tangent ...
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1answer
82 views

About timelike surfaces with non-diagonalizable shape operator.

Context: Consider the Lorentz-Minkowski space $\Bbb L^3 = (\Bbb R^3, {\rm d}s^2)$, with $${\rm d}s^2 = {\rm d}x^2+{\rm d}y^2 - {\rm d}z^2.$$ Take a differentiable surface $M \subset \Bbb L^3$, ...
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1answer
36 views

parallel vector field

I was wondering about the following: I know that a vector field along a geodesic that is parallel has a constant angle to the tangent vector of the curve and constant length. Now, is the converse ...
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1answer
36 views

Definition of covariant derivative of a covariant derivative

If we have a connection $\nabla$ different than the Levi-Civita connection, and for a Riemannian metric $g$ and $\nabla$ this relation is valid: \begin{align} ...
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1answer
69 views

Are $T\mathbb{S}_2$ and $\mathbb{S}_2 \times \mathbb{R}^2$ different?

I have seen the claim that $T\mathbb{S}_2$ and $\mathbb{S}_2 \times \mathbb{R}^2$ are not diffeomorphic, but I have only ever seen the proof that they are not isomorphic as vector bundles (which is a ...
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2answers
59 views

Is the shortest path in flat hyperbolic space straight relative to Euclidean space?

I have the following metric $$ ds^2 = dt^2-dx^2 $$ and I wanted to prove to myself that the shortest path for this metric is straight. I used the following relation $x=f(t)$ and $$ S = ...
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1answer
72 views

What is the relation between $C^\infty$-linear and tensorial?

I've been self studying Riemannian Geometry through Spivak's and Lee's books, and fairly often I've seen an argument that goes somewhat like this: We have an operator that acts on vectors in the ...
5
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1answer
59 views

Existence of diffeomorphism through convergence in Hausdorff distance

I'm reading a book and have come across something that I cannot verify or fix. The assumption is that $\Omega_1, \Omega_2, ...$ is a sequence of connected open sets in $\mathbb{R}^n$ that converge in ...
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1answer
2k views

How to visualize the Gaussian curvature of a 3D surface using color?

I have a 3D surface. I want to visualize color-coded Gaussian curvature. Is there any software (e.g. MATLAB, Mathematica) which can be used for calculating and visualizing the curvature in color code ...
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2answers
36 views

What does the vertical bar mean in $ \left.\frac{\partial f}{\partial x}\right\rvert $

I want to know what the symbol '|' besides a function means. For example: $$ \left.\frac{\partial f}{\partial x}\right\rvert $$
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votes
2answers
1k views

Frenet-Serret formula proof

Prove that $$\textbf{r}''' = [s'''-\kappa^2(s')^3]\textbf{ T } + [3\kappa s's''+\kappa'(s')^2]\textbf{ N }+\kappa \hspace{1mm}\tau (s')^3\textbf{B}.$$ What is $\tau$, I can't figure that part out. ...
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1answer
139 views

What is a topology?

Having read through the mathematical definition of endowing a set with a topology I must admit that I'm still struggling to conceptualise what such a mathematical construct is. I've read articles ...
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1answer
69 views

Surface of constant mean curvature

From PDE Evans, 2nd edition: Chapter 8, Exercise 12: Assume $u$ is a smooth minimizer of the area integral $$I[w]=\int_U (1+|Dw|^2)^{1/2} \, dx,$$ subject to given boundary conditions $w=g$ on ...
2
votes
2answers
66 views

Is it mathematically correct to say that if the metric is flat/curved the *shortest* path is/not a Euclidean straight line?

Is it mathematically correct to say that if the metric is flat/curved the shortest path is/not a Euclidean straight line? I am still hesitant to make this claim, due to at least one counter example. ...
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2answers
53 views

a question on topological manifolds and what topology provides

When one talks of a topological manifold being locally homeomorphic to $\mathbb{R}^{n}$ is it meant that the topology of the manifold is locally identical to a Euclidean topology such that we can ...
3
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0answers
123 views

Are geodesic flows on surfaces with negative curvature Anosov?

I'm just going through the original book by Anosov, where he tries to proof this result. I don't quite understand it. So let $\phi_t:TM\rightarrow TM$ be a geodesic flow on a compact surface $M$ of ...
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1answer
32 views

Is it true that all $k$-submanifolds of a $m$-manifold are open subsets of some closed $k$-submanifold?

Let $M$ be a $m$-dimensional (smooth) manifold. I know that $m$-submanifolds of $M$ are exactly the open subsets of $M$. Is it true that all $k$-submanifolds of $M$ are open subsets of some closed ...
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votes
2answers
615 views

what will be the parameterization of cone

I have question, I need more idea. can any one answer my question I have tried but i didnot get full idea I know this question we have to use parameterization of cone which i donot know in this case ...
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0answers
29 views

Decomposition of acceleration into normal and tangential components

If the velocity $v=\|\mathbf{v}\|$ of a point having position $\mathbf{x}(t)$ at time $t$ is never null, then acceleration $\mathbf{a}:=\frac{d^2\mathbf{x}}{dt^2}$ can be written ...
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1answer
29 views

The last coordinates of basis vectors are a chart: mistake in this example?

While trying to understand local chart on Grassmannians I came across this example in this book: Take $V = \mathbb R^2$ and $U,W$ two subspaces generated by linearly independent vectors. The books ...
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1answer
33 views

Riemannian tensor and Levi Civita connection

For a riemannian metric $g$ consider the following tensor $T_{rstu}=k(x)g_{rt}g_{su}-k(x)g_{st}g_{ru}$. Which condition has to satisfy $k$ if we want the tensor $T$ to be the Riemann tensor of a ...
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1answer
41 views

The last coordinates of basis vectors are a chart

Let $Gr$ denote the Grassmannian and let $Gr(2, T\mathbb R^3) = \bigcup_{x \in T\mathbb R^3} Gr(2, T_x \mathbb R^3)$. Consider one $2$-dimensional subspace of $\mathbb R^3$, that is, one element of ...
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0answers
49 views

Covariant derivative and box operator commutator

I know that the commutator of two covariant derivatives is giving some Riemann tensors as follow: ...
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1answer
15 views

Tangent space to noncompact Stiefel manifold

The noncompact Stiefel manifold is the set of $\mathbb{R}^{n \times p}$ matrices ($p \leq n$) that have rank $p$ (full rank). Based on my readings of ...
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2answers
27 views

Pushforward injective

Let $f : M \rightarrow N$ be a smooth surjective map between smooth manifolds. Now, consider a 2-form $\omega$ on $T_pN$. Does it now follow that the pullback satisfies? $f^* d \omega =0 \Rightarrow ...
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22 views

Deriving a curvature tensor for a special connection

May be $e(x)$ a frame vector at the point $x \in M$ for a manifold $M$. Given the following connection equation: $d_Xe+e_{|X_+}-e_{|X_-}= d_Xe + J_Xe = 0$. Here, $d_X$ denotes the directional ...
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38 views

Is there some relationship between algebraic curves and partial differential equations that goes beyond classifying different PDE's

I ask primarily because despite not having taken that many math classes (up to two semesters of a PDE class in college), it would be very interesting if maybe we could gain intuition regarding ...