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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Handling Degenerate Planes in Pseudo-Riemannian Geometry: Impact on Sectional Curvature and Comparison Theorems

I've been studying Riemannian and pseudo-Riemannian manifolds and came across an intriguing point regarding the definition of sectional curvature in both geometries. In pseudo-Riemannian geometry, for ...
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Regularity of exponential map for $C^{2,\alpha}$ Riemannian metrics

Let $g$ be a $C^{2,\alpha}$ Riemannian metric and $0<\alpha<1$. Would the exponential map $\mathrm{exp}_p$ be $C^{1,\alpha}$ as the point $p$ varies? Since $\mathrm{exp}_p$ is defined by the ...
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Regularity and rigidity of stable/unstable distribution for geodesic flow on noncompact negatively curved manifolds

For a volume-preserving $C^\infty$ Anosov flow on a three-dimensional compact Riemannian manifold, it was shown by Hurder & Katok that the Anosov foliations are always of class $C^{1, \alpha}$. ...
Guangqiu Liang's user avatar
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Construct compact submanifold containing non-compact Nash embedded submanifold

$$ \newcommand{\R}{\mathbb{R}} \newcommand{\geu}{g_{\text{Eu}}} \newcommand{\X}{\mathcal{X}} \newcommand{\iX}{\mathring{\X}}$$ Let $\X$ be a closed bounded convex set in some Euclidean space. Its ...
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What $1$-forms $\theta$ solve $\Delta \theta = f\theta$ for a smooth function $f$?

I have a seemingly basic question that I cannot find any literature on. Let $(M,g)$ be a smooth Riemannian manifold and let $\Delta:\Omega^1(M) \to \Omega^1(M)$ be the Laplace-De Rham operator on $1$-...
Julian Chaidez's user avatar
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Metric of negative curvature on connected sum

Let $(M_1,g_1)$ and $(M_2,g_2)$ be two Riemannian manifolds of dimension $n\geq 2$. If we consider the connected sum $M=M_1\mathbin{\#}M_2$ of the two manifolds; can one get a smooth metric $g$ on $M$ ...
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Growth/Decay of conformal Killing fields in cone metrics

Let $\gamma$ be a smooth metric on $S^2$ of positive curvature. Consider the metric $$g= dr^2 + r^2 \gamma$$ on $[1,\infty) \times S^2$. Does there exist a nontrivial conformal Killing field vanishing ...
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For a closed Riemannian manifold $M$, must the set of points with non-unique closest points to a closed submanifold $S$ of $M$ be of 0 volume measure?

Let $(M,g)$ be a closed (compact without boundary) Riemannian manifold of finite dimension, with the volume measure $\mu:= \mu(E):=\int_{E}dvol_g \forall E \in \mathcal{B}(M),$ the Borel sigma algebra ...
Learning math's user avatar
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Total curvature of a conjugate minimal surface

Let $s: S \to \Bbb R^3$ be an immersed minimal surface with finite total curvature and a proper annular end (possibly with other types of ends). What is exactly meant by a proper annular end? It is an ...
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Frobenius theorem and the size of integral manifold

Let $X =(X_0,X_1)\in \mathbb{R}^2$ and $Y=(Y_0,Y_1)\in \mathbb{R}^2$ be two vector fields on $\mathbb{R}^2$ such that $X,Y$ are independent on each tangent plane and $[X,Y]:=XY-YX=0$. Then by ...
George's user avatar
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Bochner Laplacian in coordinates

Sorry if this is a too basic question, but I didn't find an answer anywhere: The connection Laplacian, or Bochner Laplacian, is the differential operator acting on $k$-tensor fields $T\in\Gamma^{\...
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For riemannian manifolds, how close can a mapping from atlas be to an isometry?

Let $(M, g)$ be an $n$-dimensional $C^k$ (or $C^\infty$) Riemannian manifold. On $M$ we can define metric $d_g$ as the infimum of lengths of curves that connect given two points. Fix $x \in M$ and $r&...
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Smoothness of the Fréchet Function

Let $M$ be a compact Riemannian manifold and $d$ be the induced distance function. Suppose $\mu$ is a probability measure on $M$ with continuous density. The Fréchet function is defined as $$ F(x) = \...
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On the linearized evolution equations in general relativity

The following puzzles me already for quite some time: In mathematical relativity, especially in the discussion of the Cauchy problem, one usually works in the so-called ADM-Formalism, in which one ...
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Riemannian structure on connected Hilbert manifolds

The infinite-dimensional separable Hilbert space $H$ has the unusual property that it is diffeomorphic to its unit sphere $S^{\infty}$. Therefore, $H$ admits the round metric as a complete and bounded ...
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Simple curves on hyperbolic tori

In the paper "Simple curves on hyperbolic tori" by McShane and Rivin, they show that if $T$ is a hyperbolic once punctured torus, one can define a norm on the homology $H_1(T,\mathbb{Z})$ by ...
stupid_question_bot's user avatar
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Can every surface be realized as a mean convex hypersurface in $\mathbb{R}^3$?

I'm wondering if every closed surface can be realized as a mean convex hypersurface in $\mathbb{R}^3$, i.e. the mean curvature vanishes or points inward. Categorizing by genus: for $S^2$ ($g = 0$) ...
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Does every Spin$(7)$-manifold has a unit-length spinor?

Say $M$ be a manifold with a Spin$(7)$-structure. $M$ is spin and hence spin$^c$. Say $S=S_+\oplus S_-$ be a spin$^c$-bundle on $M$. Does $S_+$ has a nowhere vanishing section? The result is true if ...
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Connectedness of fibers of almost Riemannian submersions

EDIT: Let $M,N$ be compact connected smooth Riemannian manifolds. Let us assume that $N$ is closed, while $M$ might have a geodesically convex boundary. Given $f\colon M\to N$ be an $\varepsilon$-...
asv's user avatar
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7 votes
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Conformal Killing fields satisfy a third order PDE

Let $(M,g)$ be a $n$-dimensional Riemannian manifold with smooth metric $g$. Proposition 3.2 in the paper "The Boost Problem in General Relativity" by O'Murchadha and Chistodoulou claims ...
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Spin(7)-instanton

Let $M$ be a Spin$(7)$-manifold with a spin-bundle $S=S_+\oplus S_-$. There's an obvious connection on $S$ which comes from lifting the Levi-Civita connection. And it induces a connection on the ...
Partha's user avatar
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How to get to the earliest time zone?

You are in a plane at some point on Earth. You want to be at the earliest time zone possible at the end of your flight. What is the optimal path to take? Formally, fix spherical coordinates $(\theta, \...
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Parallel transport on a vector bundle : expansion of the correspondance between normal and tubular coordinates

Let $(M, g^{TM})$ be a Riemannian manifold of dimension $n$. Let $X \subset M$ be a submanifold of dimension $n$ with boundary $\partial X$. Then we have a splitting of the tangent bundle $$TM \vert_{\...
hseldon39's user avatar
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When "$(\varepsilon,\delta)$-geodesic" cannot be a loop?

EDIT: Let $M$ be a smooth compact Riemannian manifold. Let $\varepsilon,\delta>0$. I call a smooth curve $\gamma\colon [a,b]\to M$ an $(\varepsilon,\delta)$-geodesic if for any $t_1<t_2<t_1+\...
asv's user avatar
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Does anyone recognize this condition on a Riemannian metric on a vector space?

In the course of studying some oscillatory integral problems, the following strange condition came up. Let $V$ be a finite-dimensional real vector space. Let us say that a smooth Riemannian metric $...
Terry Tao's user avatar
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"Almost geodesics" in Riemannian manifolds which cannot be loops

Let $M,N$ be smooth Riemannian manifolds of the same dimension. Let $0<\varepsilon<\frac{inj(N)}{100}$. Let $f\colon M\to N$ be a smooth map such that for any $x\in M$ and any $v\in T_xM$ one ...
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Asymptotics on the number of diffeomorphism classes in the Cheeger finiteness theorem

A result of Cheeger says that, given any even dimension and any $\delta > 0$, there are only a finite number of diffeomorphism types of compact simply-connected manifolds of that dimension which ...
macbeth's user avatar
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7 votes
2 answers
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Elliptic regularity on manifolds: Is this true?

Let $(M,g)$ be a Riemannian manifold (without boundary) and denote by $\Delta_{g}$ the Laplace-Beltrami operator (or any other elliptic operator if you wish). I was trying to find a reference for the ...
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Nested convex hulls in Hadamard manifold

Let $F$ be a finite set in a Hadamard manifold $H$, and $W\supset F$ is its neighborhood. Is it true that the closure of the convex hull of $F$ lies in the interior of the convex hull of $W$? ...
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Statistical invariants of Riemannian manifolds

$\DeclareMathOperator\diam{diam}\DeclareMathOperator\rad{rad}\DeclareMathOperator\iso{iso}\DeclareMathOperator\com{com}\DeclareMathOperator\con{con}$A cheap way of defining invariants of Riemannian ...
Alex's user avatar
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1 answer
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Is the heat kernel of a manifold $p$-integrable?

If $M$ is a separable, oriented Riemannian manifold, without any other assumption on its geometry, and $h$ is its heat kernel, it is known that $h(t,x,\cdot)$ is both integrable, and square-integrable ...
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Assumptions for uniform measure of SDE on manifolds

Suppose we're working on a compact, Riemannian manifold $M$. Suppose $dX_t = -b(X_t, t)\,dt + \sigma^2 \,dB_t$ is started at the uniform measure on $M$. What kind of assumptions on $b$ make it so that ...
optimal_transport_fan's user avatar
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2 answers
190 views

A triangle comparison in CAT(0) spaces

Let $pxy$ be a triangle in a CAT(0) space $X$, and $p' x' y'$ be a triangle in $\mathbf{R}^2$ such that the lengths $|px|=|p'x'|$, $|py|=|p'y'|$ and the angle $\angle(xpy)=\angle(x'p'y')$. Let $z\in ...
Mohammad Ghomi's user avatar
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0 answers
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Harmonic maps of Euclidean space to the sphere

What are some examples of non-constant harmonic maps from Euclidean 3-space with its usual metric, punctured at a discrete set of points, to the 2-sphere with its usual metric?
Ben McKay's user avatar
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Does the Hodge star operator determine the metric?

Let $M$ be a closed oriented smooth $4$-manifold with two metrics $g$ and $\tilde g$. Consider the Hodge star operators on $2$-forms $$ \star:\Omega^2(M) \to \Omega^2(M) \quad \text{and} \quad \tilde \...
Zhiqiang's user avatar
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Is the geodesic flow on a Riemannian manifold conservative?

Let's consider a complete Riemannian manifold $\mathcal{M}$. The geodesic flow of $\mathcal{M}$ is a first-order flow on the tangent bundle $T\mathcal{M}$. My question: Is it conservative? By ...
Gaussioa's user avatar
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Perturbation of the self-dual harmonic $2$-forms

Let $M$ be a closed oriented smooth $4$-manifold with $m=b^+_2(M)>0$. We set $\mathcal R$ to be the space of smooth metrics on $M$ and consider the following map $f:\mathcal R \to Gr(m,n)$ defined ...
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Parallel transport of global sections and Riemannian curvature

A, perhaps, naive question from an algebraist/combinatorialist teaching differential geometry. Originally asked on math.SE but didn't receive a single comment in 3 days. Consider a (real) smooth ...
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Measurability of the union of cut loci along a curve

Let $(M,g)$ be a Riemannian symmetric space and $\alpha(s)$ be a geodesic. Define $$ U(t)=\cup_{s\in[0,t]}{\rm Cut}(\alpha(s)) $$ as the union of the cut loci ${\rm Cut}(\alpha(s))$ along the curve $\...
Hengchao Chen's user avatar
5 votes
2 answers
473 views

How to learn intrinsic torsion

I want to learn about G-structure and intrinsic torsion. But I can find no textbook that details it. If you can give me a reference about it, it would be much appreciated.
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1 answer
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Osculating sphere at point of maximal curvature lies to one side

I'm looking for the higher dimensional version of this post, which says that given a curve $\gamma \subseteq \mathbb{R}^2$, the osculating circle will lie to one side of the curve at points of maximal ...
JMK's user avatar
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1 vote
1 answer
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Differentiability of an integral of geodesic distance

Let $(M,g)$ be an $m$-dimensional Riemannian symmetric space and $d(\cdot,\cdot)$ be the geodesic distance function. Fix any $\alpha\in M$ and $v\in T_\alpha M$ with $\|v\|=1$. Q1: Define $$ g(t)=\...
Hengchao Chen's user avatar
5 votes
1 answer
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Clarifying a result of Klingenberg

I am looking for some clarification on a result of Klingenberg. For context, I have a complete Riemannian manifold for which I would like to compute a lower bound on the injectivity radius at each ...
E G's user avatar
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$L^{p}$ estimate for $\Delta|\nabla u|$ on a manifold with bounded Ricci curvature

This is a more advanced question than Estimates of $\Delta|\nabla u|$ for harmonic function $u$ . The Bochner formula and refined Kato inequality tells us that on a Riemannian manifold $(M^{n},g)$, if ...
Xin Qian's user avatar
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7 votes
1 answer
298 views

Estimates of $\Delta|\nabla u|$ for harmonic function $u$

The well-known Bochner formula and refined Kato inequality (for harmonic function) tells us that for a harmonic function $u$ in $B_{2}\subset\mathbb{R}^{n}$, $$ \frac{1}{n}|\nabla^{2}u|^{2}\leqslant |\...
Xin Qian's user avatar
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10 votes
3 answers
2k views

Is there an absolute geometry that underlies spherical, Euclidean and hyperbolic geometry?

A space form is defined as a complete Riemannian manifold with constant sectional curvature. Fixing the curvature to +1, 0 & -1 and then taking the universal cover by the Killing–Hopf theorem ...
Mozibur Ullah's user avatar
18 votes
1 answer
999 views

What is the largest subset of the sphere such that inner product of any two points in the set is nonnegative

I'm interested in the question of finding the maximum area of $A\subset S^{d-1}$, such that, for all $x,y \in A, \left<x,y\right>\ge 0$. The portion of the sphere lying in the positive orthant ...
RandomTensor's user avatar
1 vote
0 answers
37 views

Gradient estimate of the eigenfunction of Laplacian on hyperbolic space

I am trying to understand the asymptotic behaviors of the gradient of the eigenfunction function of the Laplace-Beltrami operator on the hyperbolic plane $\mathbb{H}^2$. Specifically, my focus lies on ...
Atlas Tasilli's user avatar
2 votes
0 answers
87 views

Convergence of diffeomorphisms

Let $(\Sigma, g)$ be a compact $n$-dimensional Riemannian manifold without boundary. Let $F_i$ be a sequence of diffeomorphisms of $\Sigma$ and $u_i$ be a sequence of positive scalar functions. ...
user486255's user avatar
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1 answer
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How to estimate the distance between geodesics and points for Riemannian manifold with positive sectional curvature

Assume that $ M $ is a complete Riemannian manifold and there exists $ k>0 $ such that $ K(q)\geq k $ for any $ q\in M $, where $ K $ is the sectional curvature of $ M $. Let $ \gamma $ be a closed ...
Luis Yanka Annalisc's user avatar

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