# 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 ...

16 views

### Independence of parametrization in defining integral of differential form

This is an exercise from Spivak's Calculus on Manifolds. Questions asks the following : (Independence of parametrization). Let $c$ be a singular $k$ cube and $p:[0,1]^k\rightarrow [0,1]^k$ be ...
40 views

### Einstein notation

I'm confused about a specific issue that I have with the Einstein notation (for tensor fields on manifolds). I want to write the following thing: Let $X$ be a smooth manifold. Choosing local ...
9 views

### Conformal curvature line parametrization

While reading a paper I found a definition which is confusing me. Def: A conformal curvature line parametrization $(x,y) \to F(x,y)$ is called isothermic. I know what a conformal ...
20 views

### moving frame with maple

I have already ask this question on stackoverflow, but since it concerns as mathematics than computer science, I ask it here too. I would like to make a classical computation using maple. I would ...
17 views

### Proof of relation between normal of a surface and principle curvatures of surface.

If F(x,y,z) is a scalar function. Then how to prove that, $$\nabla . n = K_1 +K_2$$ where n is normal to surface of constant $F$ given as $$n=\frac{\nabla F}{|\nabla F|}$$ $K_1$ and $K_2$ are ...
18 views

### Computing Gauss's of a sphere

The vector field given as $\vec{F}=\frac{\left \langle x,y,z \right \rangle}{\sqrt{x^{2}+y^{2}+z^{2}}}$ The region $D=\left \{ a^{2}\leq x^{2}+y^{2}+z^{2}\leq b^{2} \right \}$ I've some ...
55 views

### When is a diffeomorphism analytic?

I've read somewhere "a class $C^\infty$ diffeomorphism is said to be analytic" but I forgot to write down where I read this and now I can not find it, which makes me wonder if it's true? I'm working ...
34 views

### When is the metric completion of a Riemannian manifold a manifold with boundary?

Let $(M,g)$ be a connected smooth Riemannian manifold and denote by $(M,d)$ the induced metric space following by taking topological metric to be the infimum over length of curves in the standard way. ...
26 views

44 views

### What are some interesting uses for/motivations of projective spaces?

I have trouble motivating myself to think about real projective spaces, for instance. Are there any cool results about them? Are there any motivating examples?
27 views

20 views

### Hermitian Structure on $\mathcal{O}(1)$ line bundle over $\mathbb{P}^{n}$

I'm reading through Huybrecht's section on Vector Bundles. In general, he notes that for a (holomorphic) line bundle $L$ with $s_{1}, \ldots, s_{k}$ globally generating (holomorphic) sections, we can ...
30 views

### Extending automorphisms on surfaces

Assume we are in the complex setting. Let $X$ be a surface, $C$ a curve on $X$. Say $X-C$ is isomorphic to some $X'-C'$ whith $X'$ a surface and $C'$ a curve on $X'$. If it helps we may assume that ...
16 views

### Geodesics on surfaces of revolution about z axis with negative curvature

This is a question in differential geometry of surfaces that I could not do We are given S a surface of revolution about the z axis with everywhere negative Gaussian curvature. We are to show that the ...
26 views

### Why is the backwards curve needed for the inverse of a parallel transport?

Dearh math.stackeschange-community, I'm at a loss with the following problem: Let $I=(a,b)$, then the inverse of the parallel transport $P_{s,t}^\gamma$ from $s \in I$ to $t \in I$ along the curve ...