# Tagged Questions

Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.

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### Riemannian manifolds conformally diffemorphic to sphere [on hold]

let a Riemannian manifold be conformally diffeomorphic to a sphere. Is its scalar curvature constant?
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### Limit space of a sequence of Riemannian manifolds with uniformly bounded below Ricci curvature

Let $\{M^n_i\}_{i=1}^\infty$ be a sequence of closed smooth Riemannian $n$-dimensional manifolds with uniformly bounded below Ricci curvature and uniformly bounded above diameter. The Gromov ...
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### Is there any open Ricci-flat ALE 4-manifold other than Hyper-Kahler ALE 4-manifolds?

Concerning my previous question Non simply connected HyperKähler 4-manifolds without ALE metrics the following question occurred to me: Is there any open Ricci-flat ALE 4-manifold other than ...
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### The Lie algebra of Harmonic functions

Is there a Riemannian metric $g$ on $\mathbb{R}^{2}$ with corresponding volume form $\omega= \sqrt{det(g_{ij})} dx \wedge dy$ and the corresponding Laplace operator $\Delta$ such that the space ...
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### Umbilic points on Euclidean hypersurfaces

Every smooth embedding of $S^2$ into $\mathbb{R}^3$ has at least one umbilic point (in fact, the recent proof of the Caratheodory conjecture yields two such points). The usual proof of this is to use ...
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### Question on period map, Gauss-Manin connection and complex coordinates of $\mathcal{H}^1(k)$

Let $\mathcal{L}_g$ be the space of abelian differentials on Riemann surfaces of genus $g\ge 2$ and $\mathcal{TH}_g:=\mathcal{L}_g/Diff_0^+(S_g)$ be the Teichmuller space of abelian differentials on ...
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### When can a Connection Induce a Riemannian Metric for which it is the Levi-Civita Connection?

As we all know, for a Riemannian manifold $(M,g)$, there exists a unique torsion free connection $\nabla_g$, the Levi-Civita connection, that is compatible witht metric. I was wondering if one can ...
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### Twisting Spinor Bundles with Line Bundles

In a paper I am reading, the following framework was given: Let $S$ be a spinor bundle, over a Riemannian manifold $M$, with Clifford action $$c:S \otimes \Omega^1(M) \to S.$$ Moreover, let $E$ be ...
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### Chebyshev net in 3D

I would like to know the reasons why the existance of Chebyshev net in 3D-case is problematic. This question boils down to the PDE described below. (I do not know much about PDEs, so feel free to say ...
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### Does every smooth manifold admit a metric with bounded geometry and uniform growth?

Let $M$ be a smooth manifold, $g_M$ a Riemannian metric, and consider for $x\in M$ the volume growth function, $gr_x$ that maps $r>0$ to the volume $vol_{g_M}(B(x,r))$. My interest is to see ...
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### Parallel transport of a manifold logarithm

Let $x$ and $y$ denote two points on a Riemannian manifold $M$ and let $\log_xy$ denote the logarithmic map (corresponding to a given metric) applied to $y$ at $x$. Also, let $P^{x\rightarrow y}$ ...
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### k-flats in homogeneous spaces

In a symmetric space of rank $k$ (and I'll take $k > 1$) every geodesic is contained in a $k$-flat: a totally geodesic, flat, connected, and closed submanifold of dimension $k$. Question. Are ...
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### Short examples that are/are not quantum-ergodic

Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic? Note that a (compact) Riemannian manifold is said to be quantum ergodic if almost ...
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### Question on Weil-Petersson metric on Teichmuller space

I'm reading Ahlfors' original articles about Weil-Petersson metric: "Some remarks on Teichmüller's space of Riemann surfaces" and "Curvature properties of Teichmüller's space". The tangent space at ...
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### Why is it easy to compute the first and fourth moments for random chord length in a convex solid?

Recently I was led to some considerations in geometric probability, a field pretty far from any specialization of mine. (Context: I was working with a collaborator on a question about mean escape ...
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### A connection on $Hom( E,E)$ whose parallel transport is compatible to parallel transport of $E$

According to the answer of Sebastan and previous edit of Ben McKay I revise my post as follows: Assume that $E$ is a vector bundle over a manifold $M$ with a connection $\nabla$. Is there a (...
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### Recover Embedding from Metric

Suppose that $M$ is an embedded sub-manifold of $D$-dimensional Euclidean space $E^D$, with embedding $\phi:M \hookrightarrow E^D$; the embedding is not explicitly known. And suppose that I know ...
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### When Spectrum of Laplacian in a non-compact manifold is infinite and discrete?

We know that the spectrum of Laplacian in compact smooth manifolds are discrete and infinite. There is a question about spectrum of Laplacian in non-compact case in mathoverflow, Spectrum of ...
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### Isometric embedding of SO(3) into an euclidean space

Consider $SO(3)$ with its bi-invariant metric and $R^n$ the euclidean space of dimension $n$. What is the minimal value of $n$ such that there exists an isometric embedding $f: SO(3) \to R^n$?
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### Is a G-invariant metric always Kähler-Einstein?

Suppose there is a Hermitian symmetric space of compact type $X$. It is realized in the following way: $X\hookrightarrow\mathbb{P}^N$ and equipped with the induced Fubini-Study metric $g$. What's ...
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### asymptotic behavior of Lipschitz constants of sectional curvature

I'm studying the paper "Measure Concentration and the Topology of Positively-Curved Riemannian Manifolds" (https://arxiv.org/pdf/1402.4947v1.pdf) and I have some problem in understanding the proof of ...
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### Reference: Finsler Derivative?

On the wikipedia page "Generalizations of derivative" the author mentions: " in Finsler geometry, one studies spaces which look locally like Banach spaces. Thus one might want a derivative with some ...
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### How do spectrums interact with bi-Lipschitz maps?

If it makes things simple, we can just stick to bi-Lipschitz maps from $S^k \rightarrow \mathbb{R}^d$ (w.r.t geodesic distance on the sphere with the standard round metric and the $2-$norm on the ...
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### Counting limit cycles via curvature in Riemannian geometry

In this post we would like to give a possible new approach to the second part of the Hilbert 16th problem First we give a short introduction: A quadratic system is a polynomial vector field on ...
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### A Geometric proof of the Gauss Lucas theorem

Motivated by a geometric proof of the Fundamental Theorem of Algebra we ask: Is there a geometric proof for the Gauss-Lucas theorem ?Since we are working on a half plane, can one imagine a possible ...
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### Algebraic (semi-) Riemannian geometry ?

I hope these are not to vague questions for MO. Is there an analog of the concept of a Riemannian metric, in algebraic geometry? Of course, transporting things literally from the differential ...
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### Clarification on Étienne Ghys' “Feuilletages riemanniens sur les variétés simplement connexes” paper

I apologize for this type of question, but I'm having some trouble to understand remark 3.4(4) on page 212 of this article, that reads The restriction of $\overline{\mathcal{G}}$ (the foliation ...
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### Ricci curvature and killing form

Motivated by this question we ask: Is there any relation between the Ricci curvature of a Lie group and the killing form of its Lie algebra?Under what conditions, they are proportional to each ...
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### sequence of graphs converge in the sense of varifold to multiplicity 2 plane

Say in $R^3$, is there a sequence of smooth graphs $f_n$ over some plane P, such that the graphs as submanifolds in $R^3$ converge in the sense of varifold (as Radon measures on $R^3 \times Gr(2,3)$ ) ...
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### Curvature of a Lie group

Since a lie group is a manifold with the structure of a continuous group, then each point of the manifold [Edit: provided we fix a metric, for example an invariant or bi-invariant one] has some scalar ...
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### Questions on J. F. Nash's answer about his errors in the proof of embedding theorem

In the interview of John Nash taken by Christian Skau and Martin Gaussen, in EMS Newsletter, September, 2015 when asked Is it true, as rumours have it, that you started to work on the embedding ...
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### Existence of a brake geodesic on a non compact Riemannian mfd

I am interested how to find a geodesic (if it exists) on a Riemann manifold s.t. the geodesic connects 2 different points on the edge of the manifold the metric is positive definite everywhere on ...
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### minimal surfaces in $S^n$

Thanks to Choi-Schoen theorem, we know that the space of embedded minimal surfaces into $S^3$ of fixed genus is compact. My question are simples: Can we remove the embeddness assumption? Can we ...
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### Are there some intrinsic invariants of surfaces other than Gaussian curvature?

The principal curvatures of a surface is denoted by $\kappa_{1}, \kappa_{2}$. Let $P(x,y)$ be a polynomial with real coefficients. Assume that $P(\kappa_{1}, \kappa_{2})$ is an intrinsically ...
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### $W^{1, p}(M, N)$ path-connected if and only if $C^0(M, N)$ is path-connected

I'm asked to show that for compact, smooth Riemmanian manifolds $M$, $N$ we have that $W^{1,p}(M,N)$ is path-connected if and only if $C^0(M,N)$ is path-connected. The theorem (0.1) is taken from "...
Let $(M,g)$ be an $n$-dimensional, connected, compact, oriented, smooth Riemannian manifold with boundary. Assume we are given an immersion $f \colon M \to \mathbb{R}^n$ (note that $n=\dim M$). Let \$...