Tagged Questions

6
votes
1answer
52 views

Hamiltonian for Geodesic Flow

I'm trying to prove that geodesic flow on the cotangent bundle $T^* M$ is generated by the Hamiltonian vector field $X_H$ where $$H = \frac{1}{2}g^{ij}p_i p_j$$ but I am stuck. Could somebody show ...
0
votes
0answers
42 views

Geodesic equation for a 2D manifold

I am having trouble understanding how the following statement (taken from some old notes) is true: For a 2D manifold such that $$ds^2=\frac{1}{u^2}(-du^2+dv^2)$$ If we assume that $$\dot x^a\dot ...
1
vote
0answers
74 views

Energy functional

During my study on Ricci Flow I faced some functional known as enery functional. For example Einstein-Hilbert functional is called an energy functional, also in Perelman's works ...
3
votes
1answer
57 views

Is the Structure Group of a Fibre Bundle Well-Defined?

Am I right in thinking that the structure group of a fibre bundle is any group $G$ of homeomorphisms of the fibre $F$ such that all transition functions map into $G$? Or is $G$ somehow the minimal ...
4
votes
2answers
93 views

Bundle Automorphisms, Structure Groups and Gauge Groups

I am trying to get my head around the mathematical foundations of gauge theory and wanted to check that I am correct in thinking the following is true. If $E$ is a $G$-principle bundle over $M$ then ...
2
votes
3answers
148 views

Gentle introduction to fibre bundles and gauge connections

To better understand papers like this for example, which makes heavy use of fibre bundles and gauge connections to represent gauge fields, I am looking for a nice introduction to this topic. The only ...
6
votes
1answer
226 views

Hodge Star Operator

I'm trying to understand the Hodge star operation, but have come across an impasse almost immediately. I have the definition $$(\star \omega)_{a_1\dots a_{n-p}}=\frac{1}{p!}\epsilon_{a_1\dots ...
3
votes
0answers
101 views

Curvature of Hyperbolic Space

I'm trying to prove that hyperbolic space has constant sectional curvature $-1$, but keep running into difficulties. Could someone show me a way out? I've been given the metric ...
2
votes
1answer
47 views

Geodesic First Variation

I'm trying to prove that if the first variation of length vanishes then the curve $\gamma$ must be an affinely parameterised geodesic. In the following $T=\dot{\gamma}$. So I've attacked the ...
3
votes
3answers
157 views

Definition of a tensor for a manifold

While reading Nakahara's geometry, topology and physics. I came across the following definition of a tensor. A tensor $T$ of type $(p, q)$ is a multilinear map that maps $p$ dual vectors and $q$ ...
0
votes
0answers
46 views

A doubt about fuchsian functions in physics?

I'm not sure if this is the right place (or physics.stackexchange?) to ask the next What is the difference between fuchsian, theta-fuchsian, and kleinian functions? Please, suggest me an ...
0
votes
0answers
91 views

Chain rules for differential forms

I have the variable $x,y,z$ possibly depending on each other i.e. on a smooth manifold. Using the theory of differential forms I can derive $\left(\frac{\partial x}{\partial y}\right)_z ...
1
vote
1answer
52 views

bundle isomorphism

Let $M$ be a manifold and let $U_{\alpha}$ and $U_{\beta}$ be coordinate charts with coordinates $x^{\alpha}$ and $x^{\beta}$, respectively. How to show that $f_{\alpha} : ...
0
votes
1answer
110 views

Differentiable structure on torus

I defined torus as quotient space of $\mathbb{R}^2$, and $\pi$ is a quotient map. Then for each $ v\in\mathbb{R}^2$ I took a neighborhood $U(v)=B_{1/3}(v)$ and looked on $\pi|U:U\to \pi(U)=V$ and ...
0
votes
1answer
172 views

Vector fields on torus

Let $A=d/dx$ and $B=d/dy$ be vector fields on $\mathbb{R}^2$, prove that they induce vector fields $X$ and $Y$ on the torus $T$ by $X(f)=A(f(\pi)), Y(f)=B(f(\pi))$ where $\pi$ is quotient map from ...
11
votes
2answers
497 views

How to introduce stress tensor on manifolds?

I want to understand the type of stress tensor $\mathbf{P}$ in classical physics. Usually in physics it is said that the force $\text d \boldsymbol F$ (vector) acting on an infinitesimal area $\text ...
2
votes
1answer
126 views

Continuity equation on manifolds

Mass conservation is usually written as $$\frac{\partial \rho}{\partial t} + \operatorname{div}(\rho \boldsymbol v) = 0$$ $\rho$ is the density and $\boldsymbol v$ is the fluid velocity. My attempt ...