2
votes
0answers
24 views

Can a quaternionic Kähler manifold be NOT Kähler?

I have an explicit construction of the metric on the quaternionic Kähler manifold $$\mathcal M = \frac{Sp(1, 1)}{Sp(1) \times Sp(1)}.$$ Arranging the four real degrees of freedom into two complex ones ...
1
vote
1answer
42 views

The definition of scalar and vector concomitant of a metric

I'm reading Defrise-Carter's paper Conformal Groups and Conformally Equivalent Isometry Groups. One might find the paper at the following link: ...
1
vote
0answers
72 views

Computation of the Frenet-Serret trihedron in $\Bbb L^3$ (Lorentz-Minkowski space)

Consider $\Bbb L^3 = (\Bbb R^3, \langle , \rangle)$, with the convention $$\langle (x_1,y_1,z_1), (x_2,y_2,z_2)\rangle = x_1x_2+y_1y_2 - z_1z_2$$ and $\| v \| = \sqrt{|\langle v, v \rangle|}$. Let ...
3
votes
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 ...
0
votes
0answers
43 views

Using the Hodge theorem to decompose the metric tensor

There has been a previous discussion about concrete constructions using the Hodge theorem , Construction of Hodge decomposition Let me try to ask here about a specific case where one is trying to ...
6
votes
1answer
50 views

Cosets for lie groups

I am looking for a general way of determining cosets for $(G\times H)/H$, where $G$ and $H$ are Lie groups. For example what are the cosets $(SU(3)\times SU(2))/SU(2)$. Is there a general method of ...
1
vote
2answers
77 views

Causal character of a surface (Lorentz-Minkowski space $\mathbb{L}^3$)

I'm trying to analyze the causal character of the surface $x^2 + y^2 - z^2 = -1$ in Lorentz-Minkowski space $\mathbb{L}^3$, with the convention $\mathrm{diag[1,1,-1]}$, that is $$\langle \left(x_1, ...
5
votes
2answers
117 views

Why are 'differential operators on manifolds' differential operators?

It is clear what is meant by a differential operator on $\mathbb{R}^n$ (or some open subset). However, it is not clear to me why differential operators on smooth manifolds are defined the way they ...
1
vote
1answer
31 views

How to proof Frobenius Theorem in general?

The general Frobenius Theorem stating that Let $u_1,\dots,u_k$ be $k$ smooth linearly independent vector field on $M$. Let $$ W=\operatorname{Span}(u_1,\cdots,u_k) $$ Then $[u_i,u_j]\in W$ for ...
2
votes
0answers
49 views

Clarification on some notation and “assumptions” in page 143-144 of the book “Quantum Fields and Strings: A Course for Mathematicians, Volume 1”

I was trying to read the chapter $1$ (at page $143$) of this book Quantum Fields and Strings: A Course for Mathematicians, Volume 1 that is supposed to be an introduction to modern quantum field ...
3
votes
1answer
64 views

Connection(gauge field) in Fubini-Study metric is pull back of a connection A of line bundle $\mathcal{O}(1)$ on $\mathbb{CP}^{N-1}$

One can describe a $\mathbb{CP}^{N-1}$ manifold with a Fubini-Study metric $g^{FS}$, and there is a connection one form $v$ on it. A is connection one form(gauge field) of a line ...
1
vote
0answers
36 views

Short examples that are/are not quantum-ergodic

Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic? Note that a (compact) Riemannian manifold is said to be quantum ergodic if ...
4
votes
2answers
73 views

Geodesics Through a Singularity

A singularity on a manifold with metric is defined to be a point at which some geodesic cannot be continued through. For example in Schwarzchild spacetime, $r=0$ defines such a point. Is it the case ...
2
votes
0answers
84 views

Geometric Cauchy Problem

I'm attending a course in Symplectic Mechanics and I have some problems in understanding something written in my lecture notes. We are in the following setting: let $Q$ be a manifold (of dimension ...
2
votes
1answer
85 views

Any $1$-form (only on $S^3$?) can be uniquely written as a sum of a closed $1$-form and a coclosed $1$-form?

What is meant by saying that any $1$-form (only on $S^3$?) can be uniquely written as a sum of a closed $1$-form and a "co-closed" $1$-form? [...Since $H^1$ of $S^3$ is trivial it follows that the ...
2
votes
0answers
34 views

On the Geroch's argument

During the study of Geroch's argument to prove positive mass theorem, I faced a problem explained below: Suppose $(M,g_{\mu \nu})$ is a four dimensional Lorentzian Manifold and $\Sigma$ is a ...
0
votes
1answer
448 views

Showing Jacobi identity for Poisson Bracket

We were given the following problem: show that $[A,[B,C]] + [B,[C,A]] + [C,[A,B]] = 0$ where $[A,[B,C]]$ et cetera are Poisson brackets. As I understand it this is a poisson bracket (where ...
3
votes
1answer
78 views

motivating the conservation of symplectic area by way of general (coordinate) covariance

I'm trying to motivate why a symplectic structure captures exactly the right structure one needs to do classical mechanics. The easiest part of this story goes like this: we need a procedure for ...
2
votes
1answer
66 views

Examples of manifolds foliated by $S^2$

I have come across the Frobenius theorem in my study of GR, which for the special case of $S^2$ roughly means, that every point of a manifold with spherical symmetry can be foliated by spheres. I know ...
0
votes
1answer
71 views

What is 'target manifold'?

I saw in a lecture about O(3) sigma model something about 'target manifold', but I do not know what is it. Is there any book I could learn about that?
14
votes
1answer
205 views

Legendre Transformation of a Lagrangian in Classical Mechanics

I have some questions about the Legendre Transformation of a Lagrangian in Classical Mechanics to the Hamiltonian: We start with a Lagrangian $L(q,\dot{q})=\frac{\langle \dot{q} , \dot{q}\rangle }{2} ...
5
votes
1answer
139 views

Quotients of $S^{2n+1}$

Any sphere $S^{2n+1} = SO(2n+2)/SO(2n+1)$ can be thought to be given as the zero-set in $\mathbb{C}^{n+1}$ of the equation, $\sum_{i=1}^{n+1} \vert z_i \vert ^2 = 1$ Now say one wants to quotient it ...
4
votes
3answers
204 views

Geometry and Physics

I have to do a presentation on Geometry and Physics. I am asking it here (rather than physics.se) because I have to focus on Geometry More than Physics. The intended audience is Undergraduate Seniors ...
3
votes
0answers
71 views

Time evolution of the worldlines of 2 particles

Suppose I have a lab frame that is freely falling in a gravitational field of the Earth -- assume non-homogeneity-- and a uniform constant electric field in the vertical direction. There are 2 test ...
3
votes
1answer
131 views

A question about Moyal product

In Geometric quantization theory the Moyal product is one of main tools. We know Moyal product for the smooth functions $f$ and $g$ on $ℝ^{2n}$ takes the form $f\star g = fg + \sum_{n=1}^{\infty} ...
5
votes
1answer
128 views

Electrodynamics in general spacetime

Let $M\cong\mathbb{R}^4_1$ be the usual Minkowski spacetime. Then we can formulate electrodynamics in a Lorentz invariant way by giving the EM-field $2$-form $\mathcal{F}\in\Omega^2(M)$ and ...
6
votes
1answer
86 views

Formal Definition of Yang Mills Lagrangian

I have a question regarding the Lagrangian in non abelian gauge theory. Say, $G$ is the gauge group and $\mathfrak g$ the associated Lie algebra. The Lagrangian is often written as $$ \mathcal ...
8
votes
1answer
200 views

Maurer-Cartan 1-form

Can anyone help me with the following? Let $\rho$ be the right-invariant Maurer-Cartan 1-form $$\rho = dg\ g^{-1}$$ I want to show that the MC equation $$d\rho - \rho \wedge\rho = 0$$ holds. So ...
2
votes
0answers
16 views

Special case of the hodge decomposition theorem [duplicate]

I am trying to prove the following special case of the hodge decomposition theorem in differential geometry for a n component vector field $V_i$ in $\mathbb{R}^n$. any vector can be written as the ...
7
votes
1answer
296 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 ...
1
vote
0answers
119 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 ...
7
votes
1answer
206 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
255 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 ...
4
votes
4answers
613 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 ...
9
votes
1answer
536 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 ...
4
votes
0answers
351 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
71 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 ...
5
votes
3answers
561 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
109 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
71 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} : ...
1
vote
1answer
201 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
329 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
616 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
206 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 ...