Questions tagged [riemannian-geometry]
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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Spectral properties of the Laplace operator and topological properties
Suppose that $M$ is a closed Riemannian manifold: one can construct the so called Laplace-Beltrami operator on $M$. Its spectrum contains some information of the underlying manifold: for example its ...
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What is known about Lie groups with (strictly) positive curvature?
If we consider $G$ a compact Lie group, there is a left invariant Riemannian metric whose the sectional curvature is nonnegative (see Milnors' paper). When can we find a left invariant metric that has ...
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Spaces of Killing spinors for different orientation
Simply put, I want to understand how a change of orientation on a Riemannian spin manifold can change the space of Killing spinors.
To be more precise:
Let $M$ be a spin manifold (i.e. the first and ...
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Structure of $C^k$ ($k<\infty$)Riemannian metrics on a manifold
$M$ is a smooth manifold. It's known that if $M$ is compact, then the space of smooth Riemannian metrics has a Frechet manifold structure. For the space of $C^k$($k<\infty$) Riemannian metrics, ...
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Obtaining the metric from the mixed Ricci tensor $R^i{}_j$
In chapter 5 of the book "Einstein Manifolds", Arthur Besse discusses the possibility to find the metric $g$ when knowing the Ricci curvature tensor $Ric(g)$ ($=R_{ij}$).
But what do we know about ...
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A clarification regarding analytic perturbation of metrics and Laplacian
This question is in reference to the following Mathoverflow question and the accepted answer to it. It seems to me that it is taken for granted that if the metric $g_t$ perturbs real analytically in ...
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Prescribing an induced metric
We know that, if we have a surface $z=f(x,y)$ with Euclidean space being ambient manifold, the induced metric is as follows (in matrix form):
$$g=\begin{bmatrix}
1+\left ( \frac{\partial f(x,y)}{\...
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Frucht's type theorem for Riemann surface
Frucht's theorem is a theorem in algebraic graph theory conjectured by Dénes Kőnig in 1936 and proved by Robert Frucht in 1939. It states that every finite group is the group of symmetries of a finite ...
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classification of homogenous complex manifolds
Suppose $X$ is a complex manifold (doesn't assume it's Kahler), and it's holomorhpic automorphism group is transitive. My question is that is there any classification of those manifolds ?
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When a Riemannian manifold with boundary is an Alexandrov space?
Let $M$ be a smooth Riemannian manifold (without boundary). Let $X\subset M$ be a smooth compact submanifold with boundary, $\dim X=\dim M$.
Under what conditions $X$, equipped with the induced ...
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Compact surface with arbitrarily large eigenvalue
Consider a compact surface $M$ with genus $\gamma \geq 2$ and fix a positive real number $V$. Is it known whether it is possible to produce a metric $g$ on the surface $M$ such that $(M. g)$ has ...
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1-parameter group of a vector field
Let $(M,g)$ be a Riemannian manifold and $\nabla$ be the Levi-Civita connection of $g$ and let $X,Y$ be vector fields on $M$. If $\lbrace \phi _t \rbrace $ is the 1-parameter group of $X$ then what is ...
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Relation between harmonic vector fields and harmonic maps
Let $f:M\longrightarrow N$be a smooth map between Riemannian manifolds and $X\in \chi (M) $ be a harmonic vector field.
What are some necessary and sufficient conditions for guaranting that $f_{\...
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On Harmonic Unit Vector Fields
When we restrict the Dirichlet energy functional to the set of all unit vector fields on a compact Riemannian manifold $(M,g)$, then the critical points of this functional are satisfied in $\Delta_g X=...
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Is this distribution completely non integrable?
We consider the usual Riemannian metric on $S^{n}$. Its corresponding LC connection gives us a distribution on $TS^{n}$. Is this distribution completely nonintegrable?
In general, what type of ...
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Lamination as limit of arcs
I am reading Bonahon's notes on closed curves, in particular the part about hyperbolic laminations. In his notes Bonahon illustrates some examples as why laminations should be "limit curves" on ...
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Which surfaces admit unbounded-length simple geodesics?
Let $S$ be a surface embedded in $\mathbb{R}^3$.
A simple geodesic on $S$ is one that does not self-intersect.
Some surfaces have simple geodesics whose length exceeds any
given bound $L$. For example,...
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Is there a proof of the uniformization theorem using circle packing?
In this paper: http://www.dm.unipi.it/~benedett/rodin-sullivan.pdf
Rodin and Sullivan show that circle packings converge to the Riemann map. Later, Scharmm and He found another proof of the same ...
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Distance between quadratic forms
In notes here http://math.univ-lyon1.fr/homes-www/gille/prenotes/lens.pdf on page $2$ a formulation of distance between two positive quadratic form $[q],[q']$ is given by
$$d([q],[q'])=\frac{\sup_{x\...
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Is every open convex subset of a Riemannian manifold necessarily contractible?
Question: Is every open convex subset $C$ of a Riemannian manifold $M$, necessarily contractible?
Here by a "convex subset" I mean a set $C$ having the property that between each pair of points in $...
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Geometry of convex subsets in Alexandrov space/ Riemannian manifold
Let $X^n$ be an $n$-dimensional complete Alexandrov space with curvature bounded below (or a smooth Riemannian manifold, possibly with boundary). Let $U\subset X$ be an open dense subset with the ...
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Is it true that given any two point in $M$ if there exists an unique geodesic joining those two points, then $M \sim \mathbb{R^n}$ [closed]
This following doubt initially came to my mind while thinking the relationship between number of genus of a manifold and number of geodesic between given two points.
DOUBT: Suppose $M\subset \mathbb{...
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Change of curvature by parallel transport
If $c$ is a normal geodesic and if $e_1$ is a unit parallel vector field, then assume that for unit vector field $v,\ v\perp e_1$, $$ R(e_1,v,v,e_1)(t) \leq R(e_3,e_4,e_4,e_3)(t) \ \ast$$
for any ...
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Transition functions under harmonic coordinate
Assume $M$ is a manifold. Assume $M$ is covered by domains $B_i$ and $\phi_i: B_i\to B_1(0)\subset{\mathbb R}^n$ are harmonic coordinates.
The Laplacian operator under a harmonic coordinate
has a ...
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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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Multiplicity of Laplace eigenvalues
Disclaimer: This is a very heuristic question and I will be satisfied with heuristic insights, if rigorous and precise answers are not possible.
All the examples of closed surfaces (or higher ...
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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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Strong equivalence between intrinsic and extrinsic metrics on $GL_n^+$?
$\newcommand{\til}{\tilde}$
Lately, I have become interested in comparing intrinsic and extrinsic metrics on Riemannian manifolds.
Consider $GL_n^+$ (invertible matrices , $\det >0$) as an open ...
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Isometry group of a compact hyperbolic surface
Consider a compact surface $M$ of genus $g \geq 2$ with a metric of constant negative curvature. My question is, is it known under what sorts of sufficient conditions such a metric will have non-...
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Ricci flow and isometry group
It is known (via Kotschwar's uniqueness of backwards Ricci flows) that the isometry group of a Riemannian metric remains unchanged under the Ricci flow. But, one can easily observe that it can change ...
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List of generic properties of Riemannian metrics
I am highly interested in compiling a list of generic properties of Riemannian metrics on a (may be compact) manifold in general, or under "relatively broad" assumptions, like generic properties of ...
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Nearly length minimizing paths are close to geodesics? [closed]
Let $M$ be a Riemannian manifold which is geodesically convex.
It's known that length minimizing curves are geodesics (after a possible reparametrization).
Now fix* points $p,q \in M$
Is the ...
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Banach manifold of paths with endpoints on submanifolds
Fix a Riemannian metric on a manifold $M$. Suppose that we fix two points $x,y \in M$. We start with the space
$C^{\infty}_{\searrow}(x,y) = \left\{\gamma: \mathbb{R}\to M:\,\lim_{t\to-\infty}\...
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Laplacian Spectra on Nearly Nodal Riemann Surfaces
Consider a family of complex curves ${\mathcal C} \to {\mathbb D}$ such that the central fibre is a nodal Riemann surface while other fibres are smooth Riemann surfaces. We choose a family of ...
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Mean value operator on Riemannian manifold
Let $(M,g)$ a Riemannian manifold. Further $M$ should be a harmonic space, that is $M$ is a symmetric and simply connected space of rank 1. (Example: Spheres $S^n$)
Consider the mean value operator, ...
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Convergence of Riemannian metrics spectra
Consider a one-parameter real analytic family of metrics $g_t$ on a compact manifold $M$ converging to a metric $g$ in $C^k$-norm, for some $k$. It is known that the Laplace spectrum of $g_t$ will ...
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Geodesic balls in warped product spaces
Let $g_S$ be a Riemannian metric on the $n$-dimensional sphere $S^{n}$ and consider the space $M=(0,a)\times S^{n}$ with the warped metric
$g=dt^2+f(t)^2g_S$, where $f\colon [0,a)\to \mathbb{R}$ is a ...
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Unique Kahler-Einstein metric $g$ with $\mathrm{Ricc}(g)=-g$ when first Chern class $C_1(M)<0$: $\mathrm{Ricc}(h)=-g\,\Rightarrow\,h=cg$ for $c>0$?
On a compact Kahler manifold, let $g$ be the unique Kahler-Einstein metric with $\mathrm{Ricc}(g)=-g$, proved to exist by Yau and Aubin when the first Chern class $C_1(M)<0$.
Question: Does $g$ ...
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A question on a result of Colin de Verdière
Consider a compact connected surface $M$ of some genus $\gamma \geq 2$. A particular case of a famous result of Colin de Verdière (see Construction de laplaciens dont une partie finie du spectre est ...
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Faster (than normal) convergence of the normalized Ricci flow on surfaces
Consider a compact surface $M$ of genus $\gamma > 1$ (I am using the more usual letter "$g$" to denote metric), and the normalized Ricci flow on it. It is known that at time $t$, the scalar ...
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Is there a canonical split signature metric on $\mathbb{P}^n\times\mathbb{P}^{n\,\ast}$?
Let
$$
M:=\{(P,\pi)\mid P\not\in\pi\}\subset\mathbb{P}^n\times\mathbb{P}^{n\,\ast}
$$
be the open and dense (and as such $2n$-dimensional) subset of non-incident point-hyperplane pairs. If $P=\mathbb{...
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Is the heat kernel for the hyperbolic plane uniformly continuous in $t\in(0,\infty)$?
let $\mathbb{H}$ be the hyperbolic plane and let $k(t,x,y)$ be the associated heat kernel.
I am wondering, if for any fixed $y\in M$ and $\epsilon >0$ the function $u_t(x):=k(t,x,y)$ is continuous ...
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On Holonomy in (regular) Riemannian Foliations
Right now, I am trying to understanding the role of holonomy fields on Riemannian foliations, which lead me to the following (probably topological) groupoid:
Let $\mathcal{F}\subset M$ be a ...
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Geodesically convex neighborhood in Finsler manifolds
It is well known that every point of a Riemannian manifold $(M,g)$ possesses a fundamental system $\{U_n\}_{n\in\mathbb N}$ of geodesically convex neighborhoods. This means that every pair of points ...
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Conformal changes of metric and geodesics
Suppose $(M,g)$ is a Riemannian manifold. Let us assume that $X$ denotes a vector field in this manifold and consider the integral curves of this vector field.
Does there exist a conformal factor $c$ ...
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Embedding Riemmanian Manifold Linearly
Given a Hilbert Manifold $M$ does there exist a smooth map into some very large Hilbert space taking geodesics to straight lines?
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Quantitative version of the splitting theorem
The classical splitting theorem (Toponogov, Milka) says that if a smooth complete Riemannian manifold (more generally, Alexandrov space) $M^n$ of non-negative sectional curvature contains a line (i.e. ...
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Compact surfaces with arbitrary gaps in spectrum
Consider a sequence of positive numbers $a_n$. My question is, can we select a closed Riemann surface whose spectrum $\lambda_i$ satisfies the condition that $\lambda_{i + 1} - \lambda_i > a_i$? Of ...
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Comparing spectra of Laplacian and Schrödinger operator
Let $M$ be a closed (compact without boundary) Riemannian manifold. Is there a body of results that compares the eigenvalues of the Laplace-Beltrami operator with that of Schrödinger operators $-\...
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Convex embedding with a positivity condition
We have a $n$-dimensional hypersurface $\Sigma$ embedded in the Euclidean $(n+1)$-space $\mathbb{R}^{n+1}$. We know that $\Sigma$ is compact without boundary, convex (not necessarly strictly convex), ...
