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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What are good Morse Theory lecture notes and books?
Searching on the net I couldnt find any recent lecture/course notes on Morse Theory. I found an old set of notes (http://www.math.toronto.edu/mgualt/Morse%20Theory/mfp.pdf) by Mike Hutchings and these ...
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The contractivity of the time derivative of the heat semigroup in $L^p$ spaces
Let $M$ be a complete manifold. The heat semigroup $e^{-tL}$ is bounded on $L^p(M)$, for any $1 \leq p \leq \infty$;
see this for instance.
It seems that we can deduce the time derivative of the heat ...
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Do cotangent bundles have "bounded geometry"?
I have often heard the phrase "a manifold $M$ has bounded geometry" thrown around without ever seeing a precise definition of what this means. Apparent examples are compact manifolds and $\mathbb{R}^n$...
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Decomposition of forms in $\operatorname{SU}(4)$-manifold
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}$Let $(X,\Omega,\omega,J)$ be a manifold with an $\SU(4)$ structure. Since $\SU(4)\subset\Spin(7)$, $X$ also has a $\Spin(7)$-structure. I ...
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Understanding exterior differential systems
Let $M$ be an $n$-dimensional smooth manifold. An exterior differential system on $M$ is by definition a graded ideal $\mathcal{I}\subset \Omega^{\bullet}(M)$ in the ring $\Omega^{\bullet}(M)$ of ...
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Is any globally-hyperbolic manifold conformally equivalent to one with complete slices?
Let $(M,g)$ be a globally-hyperbolic Lorentzian manifold. By the seminal work of Geroch and Bernal-Sánchez, we know that
$$M=\mathbb{R}\times\Sigma,\,\,\,\quad g=-\beta^{2}dt^{2}+h_{t}$$
where $\Sigma$...
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Dirac operator on $\operatorname{Spin}(7)$, $G_2$ and $\operatorname{SU}(3)$ manifolds
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}$Let's take a $\Spin(7)$ manifold $M$ (the $\Spin(7)$ structure can have torsion), then the standard Dirac operator from negavtive spinors to ...
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A Lie group whose Lie algebra is equal to (the Lie algebra? of )all functions with fibrewise polynomial growth
Let $M$ be a Riemannian manifold. We denote by $\mathfrak{g}$ the space of all smooth function $f:TM\to \mathbb{R}$ with fibre wise polynomial growth. Is it a Lie algebra wrt the Poisson bracket ...
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Non-inertial frames of reference in empty space
Imagine that somebody wants to generalize special relativity to non-inertial frames of reference. For example I am going around a point and the metrics of space is non-Euclidean from my point of view. ...
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Making sense of constant scalar curvature metric horns
Suppose we have a compact oriented surface $S$ and we remove a point $p$ on it. We could consider a neighboorhood $U$ of the puncture $p$, so that the points in this neighboorhood are described by ...
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Systole of Riemann surfaces of genus $g$
In Buser and Sarnak's "On the period matrix of a Riemann surface
of large genus", we get
$$\frac4{3}\le\limsup_{g\rightarrow\infty}\frac{\max\{\operatorname{sys}(S)|S\in\mathcal{M}_g\}}{\log ...
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Exhaustion function with uniform controls of level sets on universal covers of compact manifolds
recently I encountered the following problem in my research. Roughly speaking, it asks if, on the universal covers of a closed Riemannian manifold, one can find exhaustion functions with uniformly ...
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Doubling theorem for Alexandrov spaces
Is there a user friendly exposition of the notion of boundary of an Alexandrov space with curvature bounded from below and of the Doubling theorem?
The only reference I am aware of is the original ...
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Geodesic distance under conformal perturbation
Let $(M,g)$ be a complete Riemannian manifold of dimension $d\ge 3$. Suppose that $g_0$ is another Riemannian metric on $M$ which is conformal to $g$; i.e. $g = e^{2u}g_0$ for some $u\in C^{\infty}(M)...
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Length spaces with continuous length functional: is this set Gromov-Hausdorff closed?
As far as I can tell, a major motivation for the study of length spaces is that they arise as Gromov-Hausdorff limits of Riemannian manifolds. Specifically,
A complete connected Riemannian manifold ...
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Laplace operators that give $S^d$ eigenvalues that are perfect squares
The Laplace-Beltrami operator on the sphere $S^d$ has eigenvalues $\{ k(k+d-1) : k \geq 0 \}$. Is there a geometrically natural Laplace operator / Laplace like operator (perhaps a Hodge Laplacian or a ...
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Isometry group of the Fubini-Study metric on complex projective spaces
Let $(\mathbb CP^n,g_{FS})$ be the complex projective space equipped with the standard Fubini-Study metric.
What is the Riemannian isometry group of $(\mathbb CP^n,g_{FS})$? It seems to me that its ...
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Hamiltonian approach to Einstein manifold theory
Let $(M,g)$ be a Riemannian manifold. So $TM$ has a natural structure of a symplectic manifold. The zero section is denoted by $Z$.
We define a Hamiltonian on $T^0 M=TM\setminus Z$ via $$...
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Riemannian submersions and associated fibre bundles
My question is as follows, it is related to the chapter of Associated Fibre Bundles from [1].
Let $(X, g_X)$ and $(Y, g_Y)$ be two smooth manifolds and let $H$ be a Lie group which acts smoothly on ...
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Reference request: Schauder estimate in the space variable for parabolic equations
Setting: Let $(M,g)$ be a compact Riemannian manifold without boundary. Let $\Delta_g$ be the Laplacian and $L=\Delta_g-\partial_t$ the heat operator. Let $0<\alpha<1$, $0<t_0<T$.
Let $$u\...
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Viscosity solutions of eikonal equation on Riemannian manifolds
It is well known that given a bounded open region $\Omega \subset \mathbb{R}^n$, the Dirichlet problem $$\lVert \nabla u \rVert = 1, \quad u|_{\partial \Omega} = 0$$
admits the unique viscosity ...
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Smooth morse theory of Riemannian distance functions
Let $(M,g)$ be a Riemannian manifold, and $p\in M$. As $R>0$ increases, the topology of the ball $B(p,R)$ changes, but the changes happen only at a Lebesgue measure zero set of $R$. For instance, ...
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Is every compact smooth Riemannian manifold bilipschitz equivalent to a finite simplicial complex?
Let $M$ be a compact smooth Riemannian manifold. Then it admits a triangulation, i.e. a finite simplicial complex $K$ which is homeomorphic to $M$. Any such simplicial complex carries a natural metric ...
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A question on Cheeger-Colding theory
I'm reading Compactification of certain Kähler manifolds with nonnegative Ricci curvature by Gang Liu recently. And I feel hard to understand a statement in the paper. Now the assumption is $(M,g)$ is ...
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Einstein structure and the quotient (group)$\frac{\operatorname{Ricc}_g}{\operatorname{Iso}_g}$
$\newcommand{\Ric}{\operatorname{Ric}}\newcommand{\Iso}{\operatorname{Iso}}$Let $(M,g)$ be a Riemannian manifold with corresponding LC connection and Ricci tensor.
Is there an obvious description of ...
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Reference request: uniformization theorem proof by Borel
This answer refers to a proof of the uniformization theorem via the PDE describing metrics of constant curvature (Liouville?) by Borel. I haven’t been able to find this reference, is anyone aware ...
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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 ...
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Lower bound of mean curvature implies that the set is subset of a given ball
If a simply connected set $\Omega\subset\mathbb{R}^n$ has $C^2$ boundary such that the mean curvature $H$ of $\partial \Omega$ satisfies:
$$H\geq 1$$
Does this imply that $\Omega\subset B_1$ after ...
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Non-orientable real algebraic three-dimensional manifolds
Smooth real algebraic hypersurfaces of even degree in $\mathbb{RP}^4$ that are maximal (i.e. that are homologically as rich as possible in the sense of the Smith-Thom inequality) are all non-...
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Sign of the permutation when I show that $\star{\star w}= (-1)^{n(n-k)} w$ for the Hodge operator
Let $w=\sum_{I} a_{I}dx_{I}$ be a $k$-form in $\Bbb R ^n$. Let us consider the Hodge operator in a combinatorial form, i.e. as an $(n-k)$ form such that
$$\star(dx_{i_{1}} \wedge \dotsb \wedge dx_{i_{...
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Convergence of conformal metrics with prescribed curvature
We know that for any function $K: \mathbb{D} \to \left[-a, -b\right]$, where $a, b > 0$, there is a unique metric $h$ on the disk $\mathbb{D}$ which is conformal to $dz^{2}$, and has curvature ...
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Finite dimensional manifolds as subspace of $\mathbb{R}^\mathbb{N}$
For embedded submanifold, specifically with ambient space being $\mathbb{R}^{n}$, there are many nice properties and results. Specifically there are many examples of matrix manifolds such as the ...
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An inequality on manifolds with non-negative Ricci curvature
Let $(M, ds^2)$ be a complete Riemannian manifold of dimension $n$ such that $\mathrm{Ric}\geq 0$. Given a Lipschitz function $f\geq 0$ with $\Delta f\leq 0$ in distribution sense. For any $p\in M$, ...
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How do we calculate the gradient of this function defined using the Riemannian logarithm on a Riemannian manifold?
We consider the following function $\psi$ on an open subset $V\subset M,$ a Riemannian manifold of dimension $m,$ so that $\exp_p:U\to V$ is a diffeomorphism with its inverse $\log_p: V\to U$. Let $v\...
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A sequence of conformal metrics with bounded negative curvatures on the disc
Let $\mathbb{D}$ denote the unit disk, and let $h_{-1}$ be the unique hyperbolic metric on $\mathbb{D}$ which is conformal to $dz^{2}$.
Take a sequence of smooth complete metrics $h_{n} = e^{\rho_{n}} ...
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Elliptic equation on differential forms
Let $\Sigma$ be an $n$-dimensional smooth closed manifold ($n\ge 3$) with a non-continuous metric $g\in W^{2,\frac{n}2}\cap L^{\infty}(\Sigma)$. Let $g'$ be a fixed smooth metric on $\Sigma$, there ...
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Bound the volume by the mean curvature of the boundary
Let $M$ be a compact compact Riemannian manifold with non-negative Ricci curvature and smooth boundary $N$. Assume that the mean curvature $H_N$ of $N$ is positive.
Question: How to determine a ...
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Einstein metrics on the tangent bundle
Let $M$ be a compact, real analytic, riemannian manifold with real analytic metric $g$. Does the tangent bundle always admit an Einstein metric ?
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Which quantity could hold its convergence under Gromov-Hausdorff convergence?
Recently I've been reading T.H.Colding's paper of Ricci curvature and volume convergence. A proof of the continuity of volume functions was given under the lower Ric bounded condition.
Having searched ...
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Simple, closed geodesics in $\mathbb{S}^3$ manifold
Lyusternik and Shnirel'man were the first to prove
Poincaré's conjecture that any Riemannian metric on $\mathbb{S}^2$ has
at least three simple (non-self-intersecting), closed geodesics.
See, e.g., p....
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$\operatorname{Hess}r$ is scalar matrix $\implies$ $M$ is isometric to the space form
I'm trying to prove the rigidity part of a theorem in my paper, which requires the use of the classical Hessian comparison theorem's rigidity part:
$$\DeclareMathOperator\sn{sn}\operatorname{Hess}r=\...
CommunityBot
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A question in $\operatorname{Spin}(7)$ geometry
$\DeclareMathOperator\Spin{Spin}$I am looking for a proof of a fact (I think it's true intuitively due to representation theory) in $\Spin(7)$ geometry. Let's take a closed $\Spin(7)$-manifold $(M^8,g)...
CommunityBot
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Rauch comparison theorem for $C^{1,1}$ metrics
If $g$ is a smooth riemannian metric on $M$ with nonpositive sectional curvature, the Rauch comparison theorem implies that $(M,d_g)$ is a negatively curved metric space (every point has a ...
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Involution of 3-sphere
Suppose that a closed geodesic $\gamma$ is the fixed-point set of an isometric involution on $(\mathbb{S}^3,g)$. Assume that sectional curvature of $g$ is at least $1$.
Is it true that $$\mathrm{...
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Convex hull and least area discs in Riemannian 3-manifolds
$\DeclareMathOperator\Conv{Conv}$Let $M$ be a complete Riemannian 3-manifold and $\gamma \subset M$ a simple closed curve that bounds a least-area disc $D$ - a disc that minimizes the area among all ...
CommunityBot
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What are the zero entropy invariant measures for an Anosov geodesic flow?
Let $M$ be the double-torus with a hyperbolic Riemannian metric. The geodesic flow on the unit tangent bundle $T^1M$ has many invariant Borel probability measures. In particular there are closed ...
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Rotation in Hyperkähler manifolds
Any Hyperkähler manifold has 3 complex structures $I_{1}, I_{2}, I_{3}$. Assume that there is an additional complex structure $J$. Can this be written as $J = aI_{1} + bI_{2} + cI_{3}$, where $(a,b,c) ...
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Good orbifold and Ricci flow with Dirichlet boundary conditions on $\Sigma$
An orbifold $\mathcal O$ is a metrizable topological space equipped with an atlas modeled on $\Bbb R^n/\Gamma, \Gamma<O(n)$ finite. Let $\Sigma$ be the singular locus i.e. points modeled on $\...
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The closure of the space of Riemannian metrics with a fixed isometry class
Let $M$ be a closed manifold, and let $\mathscr{M}$ be the space of all Riemannian metrics over $M$. It is known that this is a Fréchet manifold. Consider also $\mathscr{D}$ the diffeomorphisms group ...
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Heuristics for constrained maximal volumes in hypercubes as $n \to \infty$
It can be shown that there is a unique maximal surface of revolution with constant positive Gaussian curvature embedded in $[0,1]^3$ with a pair of antipodal points as cone points which attain the ...
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