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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Existence parallel vector fields and its effect on the topology of manifolds (Karp's Thesis)
It seems that there is no digital copy of Leon Karp's Ph.D. thesis
L. Karp, Vector fields on manifolds, Thesis, New York Univ., 1976.
on internet and his paper excerpted from his thesis is very brief ...
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Smoothness of the square of the distance function on a Riemannian manifold
Let $(M^n,g)$ be a smooth Riemannian manifold. The distance between two points is the infimum of the lengths of the curves which join the points. Consider the square of the distance function
$d^2\...
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Conformal vector field on the sphere
Let's $\mathbb{S}^d$ be the unit sphere with it's standard metric $g$. A vector field $X \in \mathfrak{X}(\mathbb{S}^d)$ is conformal if and only if there is a function $f \in C^{\infty}(\mathbb{S}^{d}...
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Geometric inequality related with convexity of the boundary
I'm new to Mathoverflow, so hopefully my question is well-posed.
My problem states as follows:
Let $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain with boundary $\partial \Omega$ , $\delta&...
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Checking that the image of a curve is not contained in a hyperplane
Let $\gamma : [0,1] \to \mathbb R^n$ be a smooth curve, $n \geq 2$. I would like to find an easy to check condition such that the image of $\gamma$ is not contained in an $n-1$ dimensional linear ...
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If a Dirichlet problem is solved by transforming into ODE (proving its radial symmetry) how can we study it on manifold?
If a Dirichlet problem (elliptic PDE, in $R^{n}$) is solved by transforming into ODE (proving its radial symmetry) how can we study it on manifold?
For example, $B$ is the unit ball in $R^{n}$, the ...
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About the liouville equation $\Delta u = - \lambda e ^{u}$ on compact manifold with dimension $>2 $
I want to ask about the liouville equation $\Delta u = - \lambda e ^{u}$ on compact manifold with dimension $>2 $.
There are many studies on this equation on Riemannian surface (dimension = 2), for ...
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Geometric invariants of a Riemannian manifold encoded in certain moment map
Let $(M,g)$ be a Riemannian manifold with isometric group $G=Iso(M,G)$. The metric gives an isomorphism between tangent and cotangent bundle of $M$. So $g$ induce a natural symplectic structure on $...
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Weitzenbock- Anti-selfdual
In "The Theory of Gauge Fields in Four Manifolds", B.Lawson proves the Bochner-Weitzenbock, for an anti-self-dual field $\Psi \in \Omega^2_-(\mathfrak{G}_E)$,where $\mathfrak{G}_E$ is the ...
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2 questions about loops and negative curvature
$(M^n,g)$ is a compact $n$ dimensional manifold of negative curvature with n>2 . let $\alpha$ be a simple closed geodesic loop in $M$ based at a point $p$
1) will the geodesic in the free homotopy ...
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Can the exponential map be used to define geodesics (and hence, generalisations of geodesics)?
Let $(M,g)$ be a (connected, paracompact, $C^{\infty}$-smooth) Riemannian manifold with Riemannian metric $g$. The exponential map is defined for each point $p \in M$ to be the map $\exp_p : T_p M \to ...
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Existence of the tubular neighborhood of uniform size
Let $(M^n,g)$ be a compact Riemannian manifold with boundary $\partial M=N$. Suppose $|Rm_g| \le C_1$ on $M$ and the second fundamental form of $N$ is bounded by $C_2$. Moreover, there exists a ...
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Elworthy’s 1982 “Stochastic Differential Equations on Manifolds” - relevant?
In 1982, D. Elworthy published “Stochastic Differential Equations on Manifolds”. Apparently, this was quite a seminal book in the field of stochastic DE’s/processes on manifolds. Is this reference ...
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Special property of flat spaces
I'm reading Eschenburg's paper Local convexity and nonnegative curvature — Gromov's proof of the sphere theorem recently. And I meet a little question. In 6.1 he said: let $M=\mathbb{R}^n$ be the ...
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What are some explicit examples of nontrivial gradient almost Ricci solitons with harmonic curvature?
A Riemannian manifold $(M, g)$ is said to be an almost Ricci soliton if there exists a complete vector field $X \in \Gamma(TM)$ and a smooth function $\lambda: M \to \mathbb{R}$ such that
$$\...
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Deformation of hyperbolic structures
Let M be a hyperbolic 3-manifold, let $g_{t}$ be a smooth family of hyperbolic metrics on M. It is known that in this case we can find a family of devlopping maps $dev_{g_{t}}:\tilde{M} \to \mathbb{H}^...
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Can a manifold have a curvature-free connection that is not torsion-free?
Suppose I have a smooth manifold with a tangent bundle, and I have a connection. If this connection is curvature-free, is it guaranteed to be torsion-free? (I am not assuming a metric, just a finite-...
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Initially horizontal geodesic is always horizontal
I am trying to prove the following. (I posted this on math.se with no success)
Let $E,B$ be Riemannian manifolds. Suppose
$\pi: E\to B$ is a Riemannian submersion.
For each $x\in E$, define $V_x E = \...
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For $f$ geodesically convex with $L$-Lipschitz-gradient on hyperbolic space, is $f(x)-f(x^*)\leq(\mathrm{const}) \cdot L r$ for all $x \in B(x^*, r)$?
$\DeclareMathOperator\dist{dist}$Setting: Let $M$ be a hyperbolic space of sectional curvature $-1$, and let $f \colon M\to \mathbb{R}$ be a $C^2$, geodesically convex function which has $L$-Lipschitz-...
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Maximum symmetry metric on $ \mathbb{C}P^n $
Let $ M $ be a compact connected manifold. The degree of symmetry of $ M $, denoted $ N(M) $, is the maximum of the dimensions of the isometry groups of all possible Riemannian structures on $ M $. ...
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Non-Kähler pseudo-Kähler manifolds
A pseudo-Kähler manifold is a complex manifold $(X, I)$ endowed with a non-degenerate closed $(1, 1)$-form $\omega$. In that case, the symmetric tensor $g(\cdot, \cdot) = \omega(\cdot, I \cdot)$ is a ...
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Length and curvature for closed curves in negatively curved spaces
In the Euclidean plane, for a closed smooth curve of length $\ell$ whose curvature is bounded above by $\epsilon$ we have the inequality
$$ \ell \ge 2\pi \epsilon^{-1} $$
which follows from the fact ...
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Curvature estimate in Hamilton's Ricci flow paper for traceless $\operatorname{Rm}$ on $4$-dimensional manifold
In dimension $4$, it is known that the curvature operator $\operatorname{Rm} : \Lambda^2(M) \to \Lambda^2(M)$ admits a block decomposition of the form $$\operatorname{Rm} = \begin{pmatrix} A & B \\...
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about codimension two foliation
Are there examples of codimension 2 foliations on closed compact 4-manifolds or 5-manifold
I am curious about examples of codimension
Are there any previous studies or lecture notes of foliation ...
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Upper bound on volume growth of area minimizers
Let $M^n$ be a complete simply connected Riemannian manifold with $\operatorname{sec}_M \leq 0$ (i.e. a Hadamard manifold) and assume that there is a constant $a \geq 0$ such that $\operatorname{sec}...
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Linearization stability condition
The following is a theorem from Fischer and Marsden's 1975's paper: Linearization stability of nonlinear PDEs.
Theorem.
Let $X, Y$ be Banach manifolds and $\Phi: X \rightarrow Y$ be $C^1$. Let $x_0 \...
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Representing homotopy classes of Kähler manifolds by harmonic maps
Let $(M,g_M)$ be a compact Kähler manifold with negative bisectional curvature. Let $\alpha : (S,g_S) \to M$ be a continuous map from a compact Riemannian manifold $(S,g_S)$.
Is $\alpha$ homotopic to ...
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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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Intersection of conical neighbourhoods on a polyhedral space
Let $P$ be a non-negatively curved (in the Alexandrov sense) polyhedral space (of dimension 3, say), $p,q\in P$ be vertices, and let $e$ be an edge connecting $p$ and $q$. Assume $e$ has cone angle $0&...
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trapped geodesics
Suppose $(M,g)$ is a compact Riemannian manifold with smooth boundary. We call a point $p \in M$ regular if there exists a geodesic of finite arc length passing through $p$ with end points on $\...
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Possible isometries of a positively curved $S^2\times S^2$
Just to put things in perspective, recall that the Hopf Conjecture asks whether $S^2\times S^2$ admits a metric of positive sectional curvature. By the work of Hsiang-Kleiner, it is known that, if $S^...
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A characterization of round sphere
Let $(M,g)$ be a $k$ dimensional compact Riemannan manifold which is isometrically embeded in $\mathbb{R}^{k+1}$. The distance arising from the Riemannian metric is denoted by $d_g$.The Euclidian ...
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Induced connection on null hypersurfaces
I will use a local coordinate formalism here, since this is related to research in general relativity, and my supervisor only tolerates local coordinate formalisms. Plus the research papers I base my ...
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Free $S^1$-action on compact homogeneous spaces
Let $M = G/K$ be a compact homogeneous space, i.e. $G$ a connected compact Lie group, and $K$ a closed subgroup inside $G$ that contains no nontrivial normal subgroup of $G$.
If $r(G) > r(K)$ (...
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Ricci Curvature in infinite dimensions?
Is there a good notion of "Ricci curvature" in infinite dimensions?
My intuitive understanding of Ricci curvature is that it is some kind of an "average" of the curvature tensor over "different ...
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Does the conformal factor satisfy some equation independent of the vector field?
Let $(M, g)$ be an $(n+1)$-dimensional (pseudo-)Riemannian manifold and for any
$X \in \mathcal{X}(M)$ define its deformation tensor by
\begin{align*}
{} ^{(X)}\pi_{ab} := (\mathcal{L}_X g)_{ab} = \...
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Shrinking a disk with fixed differential
Consider mappings $f$ from $\mathbb{R}^2$ to $\mathbb{R}^2$ with differential
\begin{align}
\mathsf{d} f= \begin{pmatrix}
\cos\psi(x) &\cos\phi(y) \\
\sin \psi(x)& \sin\phi(y)
\end{...
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Let $p : \tilde{M} \to M$ be the universal cover. Can we ever deduce curvature properties of $M$ from the curvature of $\tilde{M}$?
Let $M$ be a $C^{\infty}$-smooth, connected, paracompact manifold with universal cover $\tilde{M}$. Assume $M$ is not simply connected, so that the covering map $p : \tilde{M} \to M$ is not the ...
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Exhaustion of a noncompact manifold by domains with mean convex boundary
Let $(M,g)$ be a complete noncompact Riemannian manifold. Can we find an exhaustion $M=\bigcup_{i \ge 1} U_i$ such that each $U_i$ is a bounded domain with smooth boundary $\partial U_i$ which is mean ...
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Show that continuous maps between smooth manifolds can be approximated by smooth maps WITHOUT using Whitney's embedding theorem
As it is well-known (and for example this question shows) each continuous map between smooth manifolds is homotopically equivalent to a smooth map that can be constructed using the Whitney embedding ...
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A question on future Cauchy developement
Let us consider the Minkowski spacetime $\mathbb R^{1+2}$ equipped with the metric
$$ \eta(t,x) = -(dt)^2+ (dx^1)^2+(dx^2)^2.$$
Let $\Omega$ be a bounded simply connected domain in $\mathbb R^2$ with ...
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Hyperbolic manifolds which fiber over the circle
If $N^2$ is a closed, orientable surface of genus at least $2$, and if $\phi$ is an (orientation-preserving) pseudo-Anosov mapping on $N$, then one can form the closed orientable 3-manifold $M^3$ by ...
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Convexity in co-ordinate charts of geodesic balls
Let $g_{ij}$ be a Riemannian metric tensor on an open subset $U\subseteq \mathbb{R}^n$, and let $p\in U$.
I would guess the following is true:
for $\epsilon$ sufficiently small, the $g$-geodesic ...
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Riemannian surfaces with an explicit distance function?
I'm looking for explicit examples of Riemannian surfaces (two-dimensional Riemannian manifolds $(M,g)$) for which the distance function d(x,y) can be given explicitly in terms of local coordinates of ...
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Weyl tensor of a Riemannian metric $g$
Does Weyl tensor of a Riemannian metric $g$ give information about the conformally-flatness of $g$?
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The spectrum of the Hodge Laplacian on a Riemannian manifold
The Hodge Laplacian operator on differential forms on a (compact?) Riemannian manifold carries useful information about the topology of the manifold. In particular, the multiplicity of the zero ...
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Conformal Killing spinors
In general I would like to know about the significance of conformal Killing spinors (especially keeping in mind supersymmetric theories on curved space-time).
If $\epsilon$ and the $\bar{\epsilon}$ ...
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Elliptic regularity estimate with robin boundary condition in Yamabe problem on manifolds with boundary
I meet the following boundary problem in Escobar's Yamabe Problem On Manifolds With Boundary.https://projecteuclid.org/journals/journal-of-differential-geometry/volume-35/issue-1/The-Yamabe-problem-on-...
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A question about index of Dirac operator
Let $\Phi: M\to S^n$ be a map from an even-dimensional, $\dim M=n$, spin manifold $M$ with the boundary $\partial M$ to a unit sphere. And $\Phi$ is locally constant near $\partial M$. If we take a ...
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Mean curvature under conformal transformation
I'm reading this answer:Relation between mean curvature and conformal metric.
I wonder why we need to use a new orthonormal frame in the new metric to calculate the new second fundamental form?I don't ...
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