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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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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1answer
135 views

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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1answer
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Existence of solutions of $\Delta u = f$ such that $|u|_{L^\infty} < \infty$ if $|f|_{L^\infty} < \infty$

I am struggling with figuring out the details of Proposition 7.1. in the paper Curvature and Uniformization - R. Mazzeo and M. Taylor. Setting is as follows. Let $\Omega$ be a noncompact Riemann ...
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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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1answer
507 views

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 radially symmetric isoperimetric problem

A solution to the $l$-isoperimetric problem on a Riemannian surface $(M,g)$ is a smooth closed curve $\gamma \subset M$ of length $l$ which minimizes the isoperimetric constant: $$h(\gamma) = \frac{l}{...
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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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1answer
187 views

An answer to this system of PDE's

Planning of the question: Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle The isotropic almost complex structures $J_{\delta , \sigma}$ were introduced by Aguilar on the ...
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2answers
810 views

Euler characteristic, Gauss-Bonnet, and a product formula

I know very little about the Pfaffian or how it works, and I'm new at Riemannian geometry in general. But I was wondering if there is some way to make this "intuitive" argument for the fact that a ...
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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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106 views

Gravitational instantons metric (change variables)

I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 471. At this point, he does some computations and obtains a metric: $$ \gamma dz d\bar{z}+\gamma^{...
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1answer
237 views

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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1answer
64 views

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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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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5answers
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Smoothness of the closest point on a submanifold

Let $(M,g)$ be a smooth Riemannian manifold, and let $S \subseteq M$ be a compact submanifold. Assume that for each $p \in M$, there exist a unique closest point on $S$, i.e a unique point $\tilde s(...
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1answer
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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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1answer
200 views

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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44 views

Mean Value Property for harmonic functions [migrated]

I would appreciate any insights on this matter; Let us consider the mean value property for harmonic functions in three dimensional euclidean space. This suggests that the value of a harmonic ...
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1answer
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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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3answers
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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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2answers
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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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1answer
121 views

Comparison of angles in Alexandrov space

Let $X$ be a finite dimensional Alexandrov space with curvature bounded below. Let $p\in X$ be a fixed point. Is it true that for any $\varepsilon >0$ there exists $\delta>0$ such that for any $...
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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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1answer
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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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2answers
526 views

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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0answers
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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 "...
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239 views

Are harmonic mappings of Riemannian manifolds always non-singular outside a set of measure zero?

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 $...
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Example of steady Ricci soliton whith indefinite or nonpositive Ricci curvature

I am looking for example of steady Ricci soliton with indefinite or nonpositive Ricci curvature. Any help will be appreciated. Thanks!
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If $f$ is dynamically coherent, is there a unique invariant foliation tangent to $E^{c}$?

Let $f$ be a diffeomorphism of a closed manifold $M$ such that $f$ is partially hyperbolic if the tanget bundle of $M$, $TM$ splits into three invariant sub-bunbles $$ TM = E^{s} \oplus E^{c} \oplus ...
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Yamaguchi submersion theorem

Let me remind first a theorem of Yamaguchi (1996). Given $n\in \mathbb{N}, \mu_0>0$. Then there exist $\delta_n>0$ and $\epsilon_n(\mu_0)>0$ with the following property. Let $X$ be an $n$-...
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1answer
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Differentiability of geodesics in Alexandrov subspaces of Riemannian manifolds

Let $M$ be a smooth Riemannian manifold. Let $X\subset M$ be a closed path connected subset which has curvature bounded below in the sense of Alexandrov with respect to the induced intrinsic metric. ...
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Normal variation of embedded surfaces [closed]

Let $M^3$ be a complete riemannian manifold and $\Sigma ^2\subset M^3$ a embedded minimal compact surface. Consider the normal variation $\phi: \Sigma \times \Bbb{R}\to M$ given by $$\phi(p,t)=\exp_p(...
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1answer
139 views

(quasi)metric on Riemannian manifolds via Brownian Motion?

Given points $a$ and $b$ on a Riemannian manifold $M$, I would like a (quasi)metric that corresponds to some property of Brownian Motion from $a$ to $b$ (or rather, to $\epsilon$-ball $B = \{ x : |x - ...
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Splitting of totally geodesic Riemannian foliations

Let $\mathcal F$ be a non-singular Riemannian foliation on $(M,g)$ whose leaves are totally geodesic. Suppose further that the leaves are Riemannian products of irreducible manifolds $L=L_0\times ...\...
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1answer
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geodesics on $G/K$ which are not the orbits of a 1-parameter subgroup of $G$

Let $G$ be Lie group and $K \subset G$ a closed subgroup, such that there exists a $v \in T(G/K)$ whose isotropy-group $G_v$ is discrete (so iff $\dim G_v =0$). Lets assume $g$ acts properly on $T(G/K)...
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1answer
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Does a Kähler manifold always admit a complete Kähler metric?

Every smooth manifold admits a complete Riemannian metric. In fact, every Riemannian metric is conformal to a complete Riemannian metric, see this note. What about in the Kähler case? Does a ...
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4answers
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Smoothing of the distance function on a Riemannian manifold

Suppose $(M,g)$ is a complete Riemannian manifold. $p\in M$ is a fixed point. $d_{p}(X)$ is the distance function defined by $p$ on M (i.e., $d_p(x)$=the distance between $p$ and $x$). Let $\epsilon&...
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Geometrically-explicit upper bound for on-diagonal heat kernel

Let $M$ be a compact Riemannian manifold, and $K(t;z,w)$ the heat kernel associated to the usual Laplace-Beltrami operator on functions. There are results of the form $$K(t;z,z) \leq \frac{C_M}{f_z(t)...
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0answers
268 views

The “Rolle theorem” for sections of a vector bundle

1) Assume that $E\to M$ is a smooth real vector bundle and $\nabla$ is a connection. (We do not assume any metric compatibility since we do not fix a metric on $E$). Assume that ...
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470 views

Unexpected regularity of the distance from a $C^2$ submanifold

Let $\Gamma$ be a $C^2$ compact submanifold of $\mathbb{R}^n$. Consider the distance function $\delta$ from $\Gamma$. It is well known that, for sufficiently small $\varepsilon>0$, $\delta$ is $C^2$...
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The integral of torsion

I found the following * exercise (exercise *9) in page 407 of the book of do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced in the book ...
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Two questions about Li-Yau-Hamilton estimate

This question is from my question on mathematics. Picture below is from 231 page . For to prove $Q\ge 0$ on $M \times (0,T)$, $(\partial_t -\Delta)Q \ge 0$ and $Q\ge 0$ are needed to prove.But I ...
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If $X$ is a compact smooth Riemannian manifold, why don't we integrate on a fundamental domain in the universal cover? [closed]

Let $X$ be a compact connected Riemannian manifold. The metric gives a local volume form. The universal cover is orientable, and has a precompact subspace locally isometric (with the covering metric) ...
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Uniform approximation of a continuous flow by a $\mathcal{C}^1$ flow

Setup: Consider a (smooth) compact Riemannian manifold $M$, whose distance is denoted by $d$. Let $\Phi$ be a continuous flow, namely a continuous application from $\mathbb{R} \times M $ to $M$ ...
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1answer
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Horizontal distribution of a totally geodesic foliation

Let $\mathbb{M}$ be a $n+m$ dimensional manifold. Consider on $\mathbb{M}$ a rank $n$ sub-bundle $\mathcal{H}$ of the tangent bundle. We assume that $\mathcal{H}$ is endowed with a fiber wise inner ...
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1answer
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Conformal vector field on the sphere

Let's $\mathbb{S}^d$ be the unit sphere with it's standard metric $g$. A vector field $X \in \mathfrak{X}(\mathbb{S}^d)$ is conformal if and only if there is a function $f \in C^{\infty}(\mathbb{S}^{d}...
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Definition of Levi-Civita connection map and a theorem about it?

Does anyone know definition of Levi-Civita connection map that defined as $K: TTM\to TM$. and how to prove the following theorem: Theorem: If $X\in\mathfrak{X}(M)$ be a vector field over $M$ and $...