Articles of differential geometry

Manifold diffeomorphic to $\mathbb{S}^1\times\mathbb{R}$.

Let $H=\{(x, y, z)\in\mathbb{R}^3 | \dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}−\dfrac{z^2}{c^2}=1\}$. Prove that $H$ is a manifold diffeomorphic to $\mathbb{S}^1\times\mathbb{R}$. Already I tried to be spread but not how to begin the diffeomorphism

Riemannian Metric Notation

I am just being introduced to Riemannian metrics, and I am having a bit of confusion on the notation. When reading, I’ve encountered some different notation in different sources, so I want to make sure I’m understanding what the different notation means. I’m sure there are lots of misunderstandings in what follows, so I’d appreciate […]

Find all unit speed planar curves $\alpha(s)$ such that the angle between $\alpha$ and $\alpha'$ is constant

I was trying to study for an exam in differential geometry and got stuck on the following problem : Determine all the planar curves $\alpha(s)$ parametrized by arc length, such that the angle between $\alpha$ and $\alpha’$ is a constant $0<\theta<\pi$. My attempt : The curve is planar so the torsion is zero. What I […]

Lower bound on convexity radius in terms of injectivity radius (without using curvature)

Let $M$ be a complete Riemannian manifold, and let $C$ be a subset of $M$. We will say $C$ is convex if for any points $p,q \in C$, there exists a unique normal minimal geodesic $\gamma$ joining $p$ and $q$, such that $\gamma \subset C$. We will define: the convexity radius of $M$ by “the […]

Crofton's formula for regular curves

I’m trying to use Crofton’s formula to prove that $$\int \kappa\,ds \geq 2\pi$$ for all closed regular curves. What I note is that Crofton’s formula says if $C$ is the image of a regular curve on $S^2$ of length $l$, then the measure of the set of oriented circles intersected $C$, with multiplicity, is $4l$.

The number of geodesics of a complete Riemann manifold with non-positive sectional curvature

There is a theorem of Cartan which states that if $M$ is a simply connected, complete Riemann manifold, and that the sectional curvature is everywhere $\leq 0$, then any two points of M are joined by a unique geodesic. So if we consider a Riemann manifold $M$ which is not simply connected, but is complete […]

curve of constant curvature on unit sphere is planar curve?

I’ve studied differential geometry and get this question. I’d like to verify following statement. curve of constant curvature on unit sphere is planar curve I’ve struggled with Frenet-Serret frame, differentiating, differentiating, differentiating, ….. BUT I didn’t get something yet.. Could you give me some hint, please? $$ $$ Ah!! FIRST OF ALL, I’d like to […]

Immersion, embedding and category theory

The map between smooth manifolds $f:M \to N$ is called an immersion if the tangent map $Tf$ is injective for each $x \in M$. There is a corresponding notion of submersion. About embeddings I met two definitions: the first states that a map $f:M \to N$ is called an embedding if it is an immersion […]

Divergence of vector field on manifold

This is a follow-up question to the one I made here. On the wiki page, the divergence of a vector field $X$, denoted $\nabla\cdot X$, is defined as the function satisfying $\left(\nabla\cdot X\right)\text{vol}_n=L_X\text{vol}_n$. The page gives $\nabla\cdot X=\frac{1}{\sqrt{\vert g\vert}}\partial_i\left(\sqrt{\vert g\vert}X^i\right)$, where $X^i$ is the $i^{\text{th}}$ component of the vector field $X$. If I evaluate $\text{vol}_n$ […]

Equalities and inequalities for quadrilaterals in hyperbolic space

In euclidean space any quadrilateral satisfies equalities and inequalities $$a^2 + b^2 + c^2 + d^2 = p^2 + q^2 + 4x^2$$ $$a^2 + b^2 + c^2 + d^2 \ge p^2 + q^2$$ where $a,b,c,d$ are the side lenghts, $p,q$ the lenghts of the diagonals and $x$ is the distance between the midpoints of the […]