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

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 […]

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 […]

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 […]

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 $4l$.

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 […]

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 […]

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 […]

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 $\text{vol}_n$ […]

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 […]

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