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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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7 votes
2 answers
337 views

Reference request: $\operatorname{Sym}^2_0(T^*M) \simeq \Lambda_- \otimes \Lambda_+$

I am looking for proof of the "well-known" result that for a $4$-dimensional Riemannian manifold $(M, g)$, we have an isomorphism $$ \operatorname{Sym}^2_0(T^*M) \simeq \Lambda_- \otimes \...
1 vote
0 answers
48 views

Question on gamma matrices

Let $(M,g)$ be a pseudo-Riemannian spin manifold and let us denote by $S$ the spinor bundle, i.e. the associated vector bundle with respect to the spin representation. Usually, the "gamma ...
2 votes
1 answer
134 views

Existence of Kähler Metric of Bounded Geometry on the Hermitian Vector Bundle on Projective Spaces

A Riemannian manifold $(M,g)$ is said to be of bounded geometry if the Riemannian curvature tensor and its derivatives are bounded, and it has positive injectivity radius. I am working with the ...
0 votes
0 answers
33 views

Request for resources on directional derivative of the Riemannian distance function, and Berger's lemma about geodesics realizing the diameter

I've been recently interested in directional derivatives of the Riemannian distance function, and I came across this question, and its answer by Sergei Ivanov, where he stated an important result: (I ...
4 votes
0 answers
223 views
+50

A question in spin geometry in dimension 8

$\DeclareMathOperator\trace{trace}\DeclareMathOperator\End{End}\DeclareMathOperator\Trace{Trace}$This is to understand a very specific isomorphism in dimension $8$. In dimension $4$ for a spin$^c$ ...
1 vote
0 answers
44 views

What do we know about Poisson boundaries of arbitrary Riemannian manifolds?

For closed manifolds, we know that the Poisson boundary is trivial due to compactness and for radially symmetric manifolds for which diffusion is one dimensional, there are A Brief Introduction to ...
10 votes
1 answer
605 views

Is it possible to average a riemannian metric over an action and preserve curvature bounds?

Let $M$ be a finite dimensional smooth manifold endowed with a riemannian metric $g$ and a smooth action $\mu$ by a compact Lie group $G$. Averaging $g$ over $G$ defines a new metric $$g'(X,Y)=\int_Gg(...
3 votes
0 answers
51 views

Harmonic map in the homotopy class of the identity map

Eells and Sampson's existence Theorem states that if $(N, h)$ is nonpositively curved, then a given map $f : (M, h') \to (N, h)$ can be deformed into a harmonic map in its homotopy class. Here smooth ...
1 vote
2 answers
188 views

Non-compact surfaces with non-negative Gauss curvature

Is there a topological classification of non-compact complete connected 2-dimensional Riemannian manifolds with non-negative Gauss curvature?
6 votes
0 answers
58 views

Connectedness of the space of negatively curved metrics of a compact 3-manifold

Is the space of metrics of negative sectional curvature over a closed 3-manifold connected? If so, in what paper is this result stated? Note: as the Ricci flow hyperbolizes negatively curved metrics, ...
7 votes
1 answer
318 views

Example of non homogenous manifold with a finitely generated algebra of natural functions

Let $(M,g)$ be a Riemannian manifold. Let $C^{\infty}_{Nat}(M,g)$ be the $\mathbb{R}$-algebra of scalar invariants of the curvature tensor and all its higher covariant derivatives. An example of a ...
4 votes
1 answer
547 views

Question on Lorentzian geometry

I apologize in advance if this is a too basic question. Let $(M,g)$ be a Lorentzian manifold with signature convention $(-,+,\dots,+)$. Now, lets suppose $X\in\Gamma(TM)$ defines a global time-...
3 votes
1 answer
213 views

Geodesic flows and Killing fields

How well-known is the following result: Let $(M,g)$ be a closed Riemannian manifold and define $D:C^\infty(M,Sym^mT^*M)\to C^\infty(M,Sym^{m+1}T^*M)$, $D\alpha:=\mathbb S(\nabla\alpha)$ where $Sym^m$ ...
3 votes
2 answers
247 views

Morse approximation with bounded number of critical points

Let $(M^3,g)$ be a compact Riemannian 3-manifold and let $f\in C^{\infty}(M)$ be a smooth function. Does there exist a constant $k>0$ (possibly depending on $M$ and $g$) such that $f$ can be $C^2$-...
0 votes
0 answers
109 views

Calculi of pseudodifferential operators and K-theory

I am reading the thesis of Chris Kottke (https://dspace.mit.edu/bitstream/handle/1721.1/60193/681923895-MIT.pdf) and I would need some help to try to understand intuitively why he makes the choice of ...
0 votes
0 answers
29 views

Regularity of the eigenfunctions associated to perturbed laplacian on a compact manifold

Let $M$ be a closed manifold, I consider first order laplacian perturbation associated to a density $\rho \in \mathcal{C}^\infty(M)$ with $\rho > 0$ of the form : $$ \Delta_{\rho} f = \Delta f + \...
20 votes
2 answers
2k views

Sobolev and Poincaré inequalities on compact Riemannian manifolds

Let $M$ be an $n$-dimensional compact Riemannian manifold without boundary and $B(r)$ a geodesic ball of radius $r$. Then for $u\in W^{1,p}(B(r))$, the Poincare and Sobolev–Poincaré inequalities are ...
4 votes
1 answer
284 views

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 ...
4 votes
0 answers
236 views

Jacobian of exponential map

I am playing around with the coarea formula and came across the problem of finding the Jacobian of the exponential map. Let $G$ be a compact, semisimple Lie group with associated Lie algebra $\...
1 vote
1 answer
158 views

Comparison of special metrics on Riemann surfaces with the hyperbolic one

Let $X$ be a complex algebraic curve, or equivalently a compact Riemann surface, of genus $a\ge2$. Denote $\omega_1, \dots , \omega_a$ a basis of holomorphic differentials, and consider the Riemaniann ...
7 votes
2 answers
2k views

Geodesics equation on Lie groups with left invariant metrics

First of all, I am so sorry if this question is not appropriate to be here. I tried to ask something similar on Math Stack Exchange but it didn't have much attention. Any comment and I delete the ...
9 votes
1 answer
429 views

Perturbing metrics with nonpositive curvature

Let $M$ be a compact $3$-dimensional manifold diffeomorphic to a ball. Suppose that $M$ has nonpositive (sectional) curvature and its boundary $\partial M$ is convex, or even that $M$ is a Riemannian ...
3 votes
1 answer
257 views

Asymptotic parametrization for negatively curved surfaces

Let $S$ be a complete simply connected negatively curved surface immersed in Euclidean space $\textbf{R}^3$. Does there exist a parametrization $f\colon\textbf{R}^2\to\textbf{R}^3$ for $S$ such that ...
0 votes
0 answers
60 views

Tilings of $\mathbb{R}^n$ and Riemannian manifold that is uniformly locally isometric to a ball in $\mathbb{R}^n$

Suppose that we have a Riemannian manifold $(M, g)$ that is uniformly locally isometric to a ball in $\mathbb{R}^n$, that is, there exists $r > 0$ such that for every $x \in M$ ball $B(x,r)$ in $M$ ...
1 vote
0 answers
31 views

Cut locus of linear isometric action quotients

Given a compact group $G\leq \operatorname{O}(d)$ of linear isometries on $\mathbb R^d$, equip its quotient $\mathbb R^d/G$ with the canonical orbital metric. I am curious about the following. Is ...
6 votes
1 answer
604 views

When is the cut locus a finite tree?

Let $\Omega \subset \mathbf{R}^2$ be a bounded, simply connected domain, with a regular boundary, say of class $C^2$ at least. Let the cut locus $C$ of $\Omega$ be the set of points $x \in \Omega$ for ...
3 votes
1 answer
136 views

$L^\infty$-bound on Laplace-eigenfunctions

Suppose we are on a closed Riemannian manifold $M$. Any function $f\in C^\infty(M)$ may be decomposed as $$f = \sum_{j = 0}^\infty f_j\phi_j,$$ where $\phi_j\in C^\infty(M)$ are the Laplace ...
2 votes
0 answers
70 views

Laplace spectrum on $U(n)$

Consider $\psi:SU(n)\times U(1)\to U(n)$, $(w,z)\mapsto \bar{z}\cdot w$. One can show that $\psi$ serves as a projection and $SU(n)\times U(1)$ is a principal $\mathbb Z_n$-bundle over $U(n)$. Suppose ...
0 votes
0 answers
76 views

Existence solutions of the system of equations on Riemannian manifold

Is there a way to show that the following system of two equations has a solution? I don't want to find an explicit solution, but just verify its existence. $$f''(r) + \beta \coth(r) f'(r) = \rho_0 e^{-...
2 votes
0 answers
50 views

Riemannian metrics realizable as hypersurfaces both in Euclidean and spherical spaces

I am interested in smooth Riemannian metrics on $n$-sphere, $n\geq 3$, which can be imbedded isometrically both to $n+1$-dimensional Euclidean space and $n+1$-dimensional standard sphere of radius $r$....
0 votes
0 answers
62 views

Any papers on the "priori estimate approach" for Yamabe problem

I wonder if there is any paper on the "priori estimate approach" for Yamabe problem. The Yamabe problem is solving the following equation: On a $C^{\infty}$ compact Riemannian manifold $M_n$ ...
3 votes
0 answers
97 views

Filling radius of Lens spaces

This is a question concerning Gromov's filling radius, i.e., the radius of a neighborhood of a Riemannian manifold (embedded in its Banach space of $L^\infty$-functions) at which the fundamental class ...
3 votes
1 answer
243 views

Metrics of constant Gauss curvature on 2-cylinder

Let $C=S^1\times[0,1]$ be a compact cylinder. Given positive numbers $l,\lambda>0$. Is it possible to construct a smooth Riemannian metric on $C$ of constant Gauss curvature -1 such that one ...
1 vote
1 answer
144 views

An application of min-max characterization of eigenvalues

Let $(M,g_0)$ be a $n$-dimensional closed Riemannian manifold with a Riemannian covering $(\widetilde{M},\widetilde{g}_0)$. Let $$ \mathcal{V}_{ab}=\{g\colon a^2 g_0\leq g\leq b^2 g_0\}, \quad \text{...
6 votes
2 answers
390 views

Continuity of perimeter with respect to metric

Let $\Omega$ be an open set in a closed manifold, $(M^n, g)$. We can define the perimeter as $$\text{Per}_g(\Omega) = \sup\bigg\{\int_{\Omega} \text{div}_g(T) dVol_g, \; : \; T \in C^1(M, T M), \quad \...
2 votes
1 answer
172 views

Parallel Jacobi fields in a Hadamard manifold

Let $M$ be a Hadamard manifold and let $c: \mathbb{R}\rightarrow M$ be a geodesic. A Jacobi field $Y$ along $c$ is called parallel if $Y'(t) = 0$ for every $t\in \mathbb{R}$. If we assume that $M$ is ...
3 votes
0 answers
49 views

Transport map to lower dimension?

Let $S^{d-1}$ be the sphere in $\mathbb{R}^d$. Given a $C^\infty$ function $f \colon S^{d-1} \to \mathbb{R}$, define $g \colon S^{d-1} \to S^{d-1}$ as $g(x) = \exp_x(\nabla f(x))$, where $\nabla f(x)$ ...
2 votes
1 answer
326 views

Sectional curvature and injectivity radius of natural metric in cotangent bundles

In the following paper by Cielibak, Ginzburg and Kerman (arXiv link, Comm. Math. Helv. 2004 DOI link) they claim in page $3$ that the natural metric $\tilde g$ on $T^*M$ the sectional curvature is ...
4 votes
1 answer
96 views

Sequence of 2-cylinders converging to a segment in the Gromov-Hausdorff metric

Let $\{C_i\}_{i=1}^\infty$ be a sequence of (compact) 2-dimensional cylinders with smooth Riemannian metrics with Gauss curvature at least $-1$ and geodesically convex boundary (equivalently, the ...
1 vote
0 answers
117 views

Question on globally hyperbolic manifolds and coordinates

Consider a globally hyperbolic Lorentzian manifold $(M,g)$. Then, a well-known result of Bernal-Sánchez (see Theorem 1.1 in arXiv:gr-qc/0401112) states that it can globally be written as $$M=\mathbb{R}...
1 vote
1 answer
195 views

The length is bounded

Let $\Sigma$ be a surface of finite type. Let $\mathcal{S}$ be the set of non-trivial isotopy classes of simple closed curves on $\Sigma$. One denotes by $l_x(\alpha)$ the infimal length of curves in ...
20 votes
1 answer
3k views

The continuity of Injectivity radius

$\DeclareMathOperator{\InjRad}{InjRad}$ Dear all, when reading a book of M. Berger, I learned that the injectivity radius $\InjRad(x)$ on a compact Riemannian manifold depends continuously on the ...
1 vote
0 answers
30 views

Critical point of perturbed stratifiable function has no cluster point

Given a smooth function $f:\mathbb{R}^d\rightarrow\mathbb{R}$ and a smooth manifold $\mathcal{M}$. Now, consider the set $$ S(v)=\{x:0\in x\circ(\nabla_{\mathcal{M}} f(x)+N_{\mathcal{M}}(x)+v)\}. $$ ...
3 votes
1 answer
420 views

Riemannian manifold with two geodesics

If any two dinstict points in a complete Riemannian manfiold can only be joined by two different geodesics, is the Riemannian manifold isometric to round sphere?
2 votes
0 answers
70 views

Representations of unitary group on spaces of differential forms

This is a question on certain irreducible real representations of the unitary group. My main reference is Salamon's book "Riemannian geometry and holonomy groups". The unitary group $\mathrm ...
0 votes
0 answers
141 views

Area metric vs. volume area element

Assume that $B_2\subset \mathbb{C}^2$ is an unit ball and let $d\tau(z) = dV(z)/(1-|z|^2)^3$ be associated Bergman measure on $B_2$. Then for $\Omega\subset B_2$ we define the $\tau$-volume of $\Omega$...
1 vote
1 answer
279 views

On intersection of null geodesics

Let us consider a globally hyperbolic Lorentzian manifold $(M,g)$ with empty cut locus. Suppose that $p$ is a point in $M$ and consider $C^-(p)$ to be the past null cone in $M$ emanating from the ...
3 votes
1 answer
189 views

Randomly perturbed function has no accumulated critical point almost surely?

Given a smooth function $f$ and a smooth manifold $\mathcal{M}$ in $\mathbb{R}^d$, define the set $$ S(v):=\{x:{\rm Proj}_{T_x{\mathcal{M}}}(v)=\nabla_{\mathcal{M}}f(x)\}. $$ Is correct to say that $S(...
4 votes
3 answers
285 views

Bounds on the uniformization map for a metric on the 2-sphere

Suppose $m$ is a smooth Riemannian metric on $\mathbb{S}^2$, the uniformization theorem of surfaces tell us that $m$ is conformally equivalent to the standard round metric. Formally this says that ...
2 votes
1 answer
82 views

Comparison between riemannian distance of a manifold embedded in $\mathbf R^N$ and euclidean distance

Let $M$ be a closed Riemannian manifold isometrically embedded in some $\mathbf R^N$, and moreover let $d_M$ be the Riemannian distance on $M$. It is clear that for $x,y \in M$ : $$ |x-y| \leq d_M(x,y)...

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