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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Density of zero modes
Let $(M,g)$ be a compact smooth Riemannian manifold with a smooth boundary. Let $\{(\lambda_k,\phi_k)\}_{k\in\mathbb N}$ be the spectral data on $(M,g)$, namely an orthonormal basis for $L^2(M)$ ...
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Expressing the union of principal orbits as a disjoint union of global slices for proper group actions
Setup:
I was reading about slices and principal orbit theorems (Theorem 3.4.6) from these notes.
Let the Lie group $G$ act on a complete Riemannian manifold $(M,g)$ isometrically on $M$, i.e. $\phi^{*}...
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Compactification of a Cartan-Hadamard manifold
Let $X$ be a simply connected manifold with nonpositive sectional curvature. It is standard that $X$ is uniquely geodesic, i.e., for any distinct points $p$ and $q$, there is a unique geodesic ...
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Systems of (hyperbolic) 2nd order PDEs with lower order constraints
Certain surfaces in mechanics are endowed with the fundamental forms
\begin{align}
\text{I} &= \mathrm{d}u^2+\mathrm{d}v^2+2\cos\gamma\: \mathrm{d}u\: \mathrm{d}v \\
\text{II} &= \alpha\left(\...
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Does a Riemannian submersion map horizontal geodesics to geodesics, and a relevant question?
I asked this question on MSE, but I didn't receive a response yet, so I'm asking here. Apologies if the question is not exactly a research level question, but I'm having some trouble in figuring them ...
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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 $C^{1, \alpha}$. ...
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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 $1$-...
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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 ...
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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 ...
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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 ...
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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$-...
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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 ...
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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_{\...
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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+\...
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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 $...
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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 ...
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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 ...
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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 ...
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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 ...
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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?
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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
$$
\star:\Omega^2(M) \to \Omega^2(M) \quad \text{and} \quad \tilde \...
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Is the geodesic flow on a Riemannian manifold conservative?
Let's consider a complete Riemannian manifold $\mathcal{M}$. The geodesic flow of $\mathcal{M}$ is a first-order flow on the tangent bundle $T\mathcal{M}$.
My question: Is it conservative? By ...
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Perturbation of the self-dual harmonic $2$-forms
Let $M$ be a closed oriented smooth $4$-manifold with $m=b^+_2(M)>0$. We set $\mathcal R$ to be the space of smooth metrics on $M$ and consider the following map $f:\mathcal R \to Gr(m,n)$ defined ...
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Parallel transport of global sections and Riemannian curvature
A, perhaps, naive question from an algebraist/combinatorialist teaching differential geometry. Originally asked on math.SE but didn't receive a single comment in 3 days.
Consider a (real) smooth ...
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Measurability of the union of cut loci along a curve
Let $(M,g)$ be a Riemannian symmetric space and $\alpha(s)$ be a geodesic. Define
$$
U(t)=\cup_{s\in[0,t]}{\rm Cut}(\alpha(s))
$$
as the union of the cut loci ${\rm Cut}(\alpha(s))$ along the curve $\...
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How to learn intrinsic torsion
I want to learn about G-structure and intrinsic torsion. But I can find no textbook that details it.
If you can give me a reference about it, it would be much appreciated.
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Osculating sphere at point of maximal curvature lies to one side
I'm looking for the higher dimensional version of this post, which says that given a curve $\gamma \subseteq \mathbb{R}^2$, the osculating circle will lie to one side of the curve at points of maximal ...
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Differentiability of an integral of geodesic distance
Let $(M,g)$ be an $m$-dimensional Riemannian symmetric space and $d(\cdot,\cdot)$ be the geodesic distance function. Fix any $\alpha\in M$ and $v\in T_\alpha M$ with $\|v\|=1$.
Q1: Define
$$
g(t)=\...
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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 ...
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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 ...
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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 |\...