It is well-known that you can’t equip the surface of the unit sphere with a singularity-free coordinate system. Physicists have called this theorem (which is important for the theory of black holes) “you can’t comb the hair on a coconut.” What about one dimensional higher? That is, embed a $3$-surface of constant radius in Euclidean […]

I am looking for the tangent space of $SL(n,\mathbb{R}) = \{A\in\mathbb{R}^{n\times n} | \det{A}=1 \}$ (actually $n=3$) at an arbitrary $M\in SL(n,\mathbb{R})$. From now on I will omit the $\mathbb{R}$ for convenience. It is well known and easy to prove that the tangent space at the identity matrix $\mathbb{1}$, $T_1SL(n)$, is the set of all […]

My question is related to this question I am given a $C^r$ manifold $M$ and a connected subset $A$ of $M$, and a $C^r$retraction $f:M\rightarrow A$ such that $f\vert_A=id:A\rightarrow A$. Then $A$ is a $C^r$ submanifold of $M$. I know that I need to show that $f$ has constant rank in a neighbourhood $U$ (open […]

Let’s say that $M$ is a differentiable manifold of dimension $n$. (This includes that $M$ is nonempty and second countable, so that it can be embedded into some Euclidean space, and is thus metrizable.) Is it well-defined to say that $M$ has Hausdorff dimension $n$ even though it is only a metrizable topological space? I.e. […]

Show that the Cayley sextic $$γ(t) = \bigl(\cos^3 (t)\cos (3t), \cos^3 (t)\sin (3t)\bigr),\quad t \in \mathbb R,$$ is a closed curve which has exactly one self-intersection. What is its period? I can see $2π$ is its period.But for the self-intersection have to solve $γ(a)=γ(b)=p$ ?

EDIT: Note that the object I’m seeking needn’t have anything to do with water or actual containers; those are just used to convey the idea. I’m trying to find a container that, when turned with some constant angular velocity in any direction given by $\phi$ (rotational symmetry), will pour out the same amount of water […]

Is there a long time existence for the Ricci flow on compact negatively curved surfaces? I just read that the normalized Ricci flow has a long time solution converging to a metric of constant negative curvature. I was wondering what happens to the ordinary Ricci flow, or if there is any condition under which one […]

I have a question of definition: take $(M,g)$ a riemannian manifold, $\gamma:I\rightarrow M$ a smooth path in M. We define the length of $\gamma$ as follows: $$l(\gamma) = \int_I \vert\vert\gamma'(t)\vert\vert dt = \int_I \sqrt {g_{\gamma(t)}\left(\frac{d\gamma}{dt},\frac{d\gamma}{dt}\right)}dt$$ My question is how is $\frac{d\gamma}{dt}$ interpreted as a tangent vector? I have encountered this definition problem quite a few […]

I’m trying to find a smooth degree 1 map from the torus $T^2 = S^1 \times S^1$ to the $2$-sphere $S^2$. My first thought was to use the two coordinates $(\theta_1,\theta_2)$ to map onto the usual spherical polar coordinates on the sphere – however I can’t quite get that to work because it seems that […]

I’m having serious trouble finding the umbilical points of the ellipsoid represented by $$\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1, \;\;\;a,b,c\neq 0.$$ My first thought was to use the parametrization $$\mathbf{x}(u,v)=(a\sin(u)\cos(v),b\sin(u)\sin(v),c\cos(u)),$$ for $0<u<\pi$ and $0<v<2\pi$, compute the first and second fundamental forms, etc., but this is a nightmare. After doing some researching (and on the back solutions of Do Carmo) I […]

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