# Tagged Questions

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 ...

57 views

### $\omega$ is $1$-form on $S^1$.

Let $h: \mathbb{R} \to S^1$ be $h(t) = (\cos t, \sin t)$. How do I show that if $\omega$ is any $1$-form on $S^1$, then$$\int_{S^1} \omega = \int_0^{2\pi} h^*\omega?$$
86 views

90 views

### Flow of a left invariant vector field on a Lie group equipped with left-invariant metric and the group's geodesics

I think the answer to my question is known to many other people, but I'm still getting confused. Let $G$ be a (possibly infinite dimensional also) Lie group and $g$ be its Lie algebra. Consider the ...
133 views

### Eigenvalues of a Plane Curve Laplace-Beltrami Operator

Given a closed plane curve $C$, which is a one-dimentional manifold, what are the eigenvalues of Laplace-Beltrami operator defined on this curve? I know that the LB eigenvalue problem for unit ...
119 views

### The difference between the algebraic torus and the geometric torus

I know that the donut-shaped geometric object in $\mathbb{R}^3$ is homeomorphic to a square with identified opposite sites. However, while the latter has a clear symmetry between two dimensions, the ...
116 views

### Covariant derivative (or connection) of and along a curve

Let $c:[a,b] \rightarrow M$ be a curve parametrized by arc-length. Then $c': [a,b] \rightarrow TM$ such that $c'(t) \in T_{c(t)}(M).$ So essentially, I want to understand how $\nabla_{c'}c'$ is ...
97 views

### Exercise 2.3 Lee's Riemmanian Manifolds

Statement: Suppose $M\subseteq \tilde{M}$ is an embedded submanifold. a)If $f$ is any smooth function on $M$, show that $f$ can be extended to a smooth function on $\tilde{M}$ whose restriction to $M$...
92 views

### $d(x,y)^2 \le d(x,z)^2 - d(y,z)^2\color{}{+\varphi\big(x,y,z,d(x,y),d(y,z) \big)}?$

In a metric space $(M,d)$ the triangle inequality $d (x, z) \le d(x, y) + d (y, z)$ gives us's the inequalitie $$\quad d(x,y)^2 \ge d(x,z)^2 - d(y,z)^2\;\color{}{{-2\cdot d(x,y)\cdot d(y,z)}}$$ ...
15 views

### Express covariant transformation conveniently

Let $\omega = \sum_i \omega_i dx^i = \sum_{i} \nu_i dy^i$ be a 1-form in two different bases. Now, $(\omega_1,...,\omega_n)$ transform covariantly to $(\nu_1,...,\nu_n).$ My question is, can we ...
101 views

26 views

41 views

### Wedge product of Lie algebra valued differential forms [duplicate]

Let $\mathfrak{g}$ be the Lie algebra of a matrix Lie group. Furthermore, let us consider the following $\mathfrak{g}$-valued $p$-form and $\mathfrak{g}$-valued $q$-form: \begin{array}...
369 views

### I don't get the relationship between differentials, differential forms, and exterior derivatives.

I don't get the relationship between differentials, differential forms, and exterior derivatives. (Too many $d$'s getting me down!) Here are the relevant (partial) definitions from Wikipedia; ...
96 views

### Differential forms in Physics

In the context of physics, I just read about the the symplectic 2-form $\omega$ on a symplectic manifold $M$ of dimension $2n$. Unfortunately, I could not follow a few arguments. I.e. it was said ...
64 views

### Is my graph a tree?

Let M be a smooth connected manifold. G is a group act on M cocompactly and suppose there is a harmonic function $h$ on M with minimal energy.$h:\rightarrow [0,1]$ such that h is nonconstant and ...
48 views

### How to get the result in this way?about determinant [closed]

In theory of surfaces using the fact that $\vec v \in T_p(M)$=$\alpha\vec x_u+\beta\vec x_v$ and $S_p(\vec v)\times \vec v=\vec 0,$ deduce that a nonzero tangent vector can be a principal ...
I have a lot of trouble trying to explain to myself what the author did in problem 1.102 (the answer is in the link): Let $TM$ be the tangent bundle over a differentiable manifold $M$. Let $\... 1answer 65 views ### How to show the following vector bundles are equivalent? Given a smooth sub-manifold$X$of$\mathbb{R^n}$and define the diagonal in$X \times X$to be $$\triangle = \{(x,x) \mid x \in X \} \subset \mathbb{R^n}\times \mathbb{R^n}$$ and normal bundle to$\...
Let $\Omega \subset \mathbb{R}^n$ an open, bounded and connected set with a $C^2$ boundary and a function $\rho \in C^2(\mathbb{R}^n)$ such that  \Omega = \{ x \in \mathbb{R}^n : \rho(x) < 0 \},\...