# Tagged Questions

(Stub) The study of differential geometry with notions of infinitesimal distance given by a Riemannian metric. This allows us to consider questions about the shape of a manifold by studying its curvature.

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### Easy solution to Yamabe problem for surfaces

The Yamabe problem asks if, given a Riemannian manifold $(M,g_0)$, it is possible to find a conformal metric $g$ on $M$ with constant scalar curvature. I would like to know if there is some "easy" ...
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### extension of a local orthonormal frame on a hypersurface

Let $N$ be a $(n+1)$-dimensional Riemannian manifold and $M\subset N$ a Riemannian hypersurface (embedded or immersed). Let $M$ and $N$ be oriented and choose a unit normal vector field $\nu$ along ...
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For $\delta_i \in \bigwedge^{k_i}W_i^*$, $i=1,2$, the Hodge $*$-operator of $\delta_1 \otimes \delta_2$ is given by $$*(\delta_1 \otimes \delta_2)=(-1)^{k_1k_2}(*_1\delta_1) \otimes(*_2 \delta_2)$$ ...
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### What is the meaning of the symbol $\nabla^k$?

In T.Aubin's book, a course in differential geometry, he write the formula $\Delta f=-\nabla^k\nabla_kf$ on a Riemannian manifold, but he never define the symbol $\nabla^k$. It seems that the notation ...
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### 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 ...
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### Gradient of norm of embedding

Let $\varphi:(M,g,\nabla)\to\mathbb{R}^n$ be a smooth embedding of a convex hypersurface. I want to explicitly calculate $$\langle \varphi,\varphi_{\ast}(\nabla\|\varphi\|^2)\rangle.$$ In particular, ...
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### How do we check conformal equivalence of parametrized surfaces, e.g. parallel surfaces?

Suppose we have two parametrized surfaces in $\mathbb{R}^3$: $$X,Y:\mathbb{R}^2 \rightarrow \mathbb{R}^3$$ The induced metric on either surface is the pullback of the Euclidean metric $\bar g$ due ...
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### What distinguishes elliptical coordinates from polar coordinates?

I am trying to identify what characteristic distinguishes elliptical coordinates from polar coordinates. For concreteness, let's write down the expressions. Polar: $$x=r \cos(t) \\ y=r \sin(t)$$ ...
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### Connected components of the complement of a closed geodesic on a hyperbolic surface.

Let $M$ be homeomorphic to a 2-sphere with a finite number $\geq 3$ of points removed. This implies that $M$ can be equipped with a complete, finite area hyperbolic metric. I imagine $M$ as an ideal ...
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### Function whose gradient is of constant norm

Let $f:\mathbb R^n\rightarrow \mathbb R$ be a smooth function such that $\|\nabla f(x)\|=1$ for all $x\in \mathbb R^n$ and $f(0)=0$. I would like to prove that $f$ is linear. I first looked at the ...
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### Does the possibility of linear coordinate changes imply that the manifold is Euclidean?

Question. Let $M$ be a smooth manifold admitting an atlas $\mathcal{A}$ (i.e. a collection of coordinate charts that covers the whole manifold) with the following property: for every pair ...
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### Interior of a Dirichlet domain in a Riemannian manifold

Let $X\neq\varnothing$ be a complete connected Riemannian manifold. Suppose $G$ is a group of isometries of $X$, acting properly discontinuously on $X$. We assume there is a point $x_0\in X$ such that ...
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### Calculate geodesic path on matrix manifold

I have a matrix which is change with time. Let me denote it as A(t). I know t=0 it is A(0) and I know t=1 it is A(1). A is symmetric positive semi-definite matrix. What I want to do is find the ...
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### Boundedness of Riemannian curvature gradient

I'm reading a paper by Wan-Xiong Shi "Deforming the metric on complete Riemannian manifolds". And there is a statement without proof. It can be summarized as follows: Let $B(x_{0},\gamma)$ be a ...
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### Using the Hodge theorem to decompose the metric tensor

There has been a previous discussion about concrete constructions using the Hodge theorem , Construction of Hodge decomposition Let me try to ask here about a specific case where one is trying to ...
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### Is a connected compact Riemannian manifold of dimension 1 unique?

The tiles says almost everything. It is known that a connected compact topological manifold of dimension 1 is isomorphic to $S^1$. What if we replace "topological" by "riemannian"?
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### Reformating Function

Is there such a function where a ambiguous ;n-dimensional, field/space (defined by a function) is plugged in and returns a flattened field where the basic units along the function are then formatted ...
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### Connections in non-Riemannian geometry

In case of Riemannian geometry the connection $\Gamma^i_{jk}$ as is derived from the derivatives of the metric tensor $g_{ij}$ is ought to be symmetric wrt to its lower two indices. But in the case of ...
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### Extension of smooth maps at a cusp

There is a short remark in deCarmos "Riemannian Geometry" (p. 67) and I wonder about the condition that the vertex angles must be $\neq \pi$. If $s_1$ and $s_2$ are two differentiable maps on an ...
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### Do we need a metric to define plurisubharmonic functions?

There are various notions of 'harmonicity' on various manifold. Sometimes, I am counfuesed by the definitions. For real manifold, the harmonic manifold is defined by $\Delta f=0$, where $\Delta$ is ...
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### Special expression for 3-linear symmetric map $T(X,Y,Z) = \langle -Jh(X,Y),Z \rangle$ [Ejiri]

For a project in Riemannian Geometry, I have been working out the details of a paper by Ejiri (http://www.ams.org/journals/proc/1982-084-02/S0002-9939-1982-0637177-8/S0002-9939-1982-0637177-8.pdf) ...
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### Almost complex structure compatible with Levi-Civita connection of immersed submanifold?

Suppose we have Riemannian manifold $M$ isometrically immersed in a Kähler manifold $\tilde{M}$. Let $\tilde{g}$ be the metric and $\tilde{\nabla}$ the Levi-Civita connection on $\tilde{M}$ , we know ...
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### All differentiable functions on $\mathbb{S}^n$

Consider the sphere $\mathbb{S}^n$ embedded in $\mathbb{R}^{n+1}$. Let $N$ be the north pole of the sphere and $S$ the south pole. Every point on $\mathbb{S}^n \backslash \{N,S\}$ is defined uniquely ...
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### Hyperbolic (and related) structures on open unit disk

I am facing some confusion about different structures on the open unit disk $D:=\{ z \in \mathbb{C}, |z|<1 \}$. By Riemann Mapping Theorem we know there is just one complex structure on $D$, up to ...
Suppose M is a Riemannian manifold and $g, f : M \rightarrow \mathbb{R}$ are smooth functions. When is the $1$-form Hess$^f(\nabla g, -)$ closed? I'm looking for simple conditions involving f,g and ...