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 of divergence-free unit vector field in conformally rescaled euclidean metric
Question. Let $\Omega \subset \mathbf{R}^2$ be a convex polygonal domain, equipped with a Riemannian metric $g$. Under which conditions on $g$ is there a vector field $X$ in $\Omega$ with $\mathrm{div}...
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Eigenvalues of Laplacian and eigenvalues of curvature operator
Let $(M^n,g)$ be a compact Riemannian manifold (without boundary). The symmetries of the curvature $R$ of (the Levi-Civita connection associated to) $g$ allow one to realise $R$ as a self-adjoint (...
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Books and References on Geometry of Submanifold [closed]
In this semester I want to study Geometry of Submanifolds. I know Chen Bang Yen's book: Geometry of submanifolds, but it is too hard to read since its strange print. Can people recommend textbooks and/...
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Unit Killing vector fields on pseudo Riemannian manifolds
In arXiv:math/0605371, Theorem 4 on p.8, there is the following statement:
Let $X$ be a unit Killing vector field on a $n$-dimensional Riemannian manifold $M$. Then the Ricci curvature $\operatorname{...
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Closed manifolds of nonnegative curvature operator are symmetric spaces
In an online webinar, I heard (not directly) the statement that (closed) manifolds of nonnegative curvature operator $\mathcal{R}\geq 0$ are symmetric spaces. Is this a valid theorem? Any reference ...
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The relationship between the first eigenfuntions and the second eigenfuntions on sphere [closed]
Recently I considered the following question: If we give a second eigenfuntions $g$ on sphere, then can we construct a first eigenfuntions $f$ by $g$? Is there any relationship between the first ...
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Large isometry groups of Kaehler manifolds
Let $M$ be a closed simply-connected Kaehler manifold that is not isomorphic to a product of lower-dimensional Kaehler manifolds. Pick an orientation for $\mathbb{C}$; this endows $M$ with an ...
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References Request: A paper Tanno's equation
I need a paper which wrote by Tanno where he prove that a equations $f_{ijk} + k(2f_{k}g_{ij} + f_{i}g_{jk} + f_{j}g_{ik})$ can be solved if and only if on sphere. But I can not find it on Internet ...
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Intersection Grassmanian planes
I am reading a paper that used Grassmanian planes properties. In particular, they studied the intersection of Grassmanian planes; they check the intersection Grassmanian of $n-k$-planes and ...
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largest geodesic ball inside a small portion of Euclidean submanifold
Suppose that $M\subseteq\mathbb R^D$ is a compact smooth Riemannian submanifold of dimension $d$, having normal injectivity radius $\tau$. Let $x_0\in M$ be a point, and $\delta\in (0,\tau)$ ...
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Does there exist an isometry between a regular polygon and a circle?
In order to define the question in a meaningful fashion, I am referring to a smooth manifold $\mathcal{M}$ within an $\epsilon$-neighborhood of a regular polygon $\mathcal{P}$ satisfying $$\max\{\|x-p\...
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Local existence of flat metrics with degenerate singular values
It has been proved that,
If $\lambda_1,\,\lambda_2,\cdots,\lambda_n$ are real analytic functions from $\mathbb{R}^n$ to $\mathbb{R}$, such that $\lambda_i(0)\neq \lambda_j(0)$ for $i\neq j$, then ...
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Hilbert 16th problem via hyperbolic geometry
More than 16 years ago, I heard from someone that he thinks that there is a possible relation between Hilbert's 16th problem(for $n=2$) and Hyperbolic geometry. He says that a ...
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Distance Metric on a Polytope
Primary Question: Is it possible to define a distance metric on a polytope (or permutohedron in particular)? I am aware that neither is a smooth, Riemannian manifold; however, computer scientists have ...
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Asymptotics of constant mean curvature surfaces
Let $\Sigma^n \subset \mathbf{R}^{n+1}$ be a complete, properly embedded hypersurface with constant, non-zero mean curvature $H \neq 0.$
In the case where the dimension is $n = 2$, $\Sigma$ is non-...
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Quotient of $\mathbb{R}^n$ by a subgroup of $\mathrm{SO}(n)$
$\DeclareMathOperator\SO{SO}$ Let $\mathcal{M}$ be an open subset of $\mathbb{R}^n$ endowed with the Euclidean metric and $\mathcal{N}$ be a Riemannian manifold. Assume that $G$ is a Lie subgroup of $\...
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A vector field whose flow has constant singular values
$\newcommand{\tr}{\operatorname{tr}}$
$\renewcommand{\div}{\operatorname{div}}$
Let $D \subseteq \mathbb{R}^2$ be the closed unit disk. Given a vector field $X$ on $D$, let $\psi_t$ be its flow.
Does ...
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A Kazhdan-Warner type problem
Let $X$ be a compact Riemannian manifold and I am interested in the following set of equations:
\begin{align*}
\Delta f+u\cdot e^{f+\lambda}=c\\
\lambda-2f=g
\end{align*}
where $u,g$ are given real ...
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Complete Hermitian manifolds with vanishing Chern curvature
An old theorem going back to Boothby states that a compact Hermitian manifold with Chern curvature vanishing identically is a compact quotient of a complex Lie group with a left invariant metric. Are ...
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Kulkarni-Nomizu square root of the Riemann tensor
Given a Riemann tensor $Riem$, what are conditions such that $Riem=B\star B$ for some bilinear symmetric form $B$, where $\star$ is the Kulkarni-Nomizu product? It follows from the proof of ...
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Visualizing the wave operator in two dimensions
For $n\geq 1$, let $D_n$ be the Dirac operator on the spinor bundle on the $n$-dimensional sphere $S^n$. For example, $D_1$ acts on the trivial bundle $S^1\times\mathbb{C}\to S^1$, and can be ...
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Limit cycles of quadratic systems and closed geodesics(Finitness of $H(2)$)
This question is inspired by this answer to the question Finding a 1-form adapted to a smooth flow.
Assume that $V$ is a polynomial vector field of degree $2$ as follows:$$\begin{cases} x'=P(...
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Semiconcavity estimate for the squared distance on a compact Riemannian manifold
I am currently reading this paper on the Riemannian structure of the Wasserstein space over a compact Riemannian manifold (my question doesn't concern the Wasserstein metric), specifically Section 4.1,...
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Usage/Application of Raychaudhuri equation in Riemann geometry or pure maths
While going through this paper by Witten and seeing a discussion about different aspects of Raychaudhari Equation and Einstein Field Equation. I want to ask if Raychaudhari Equation find any ...
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$\DeclareMathOperator\SU{SU}$$\SU(2)\times \SU(2)$ invariant $\SU(3)$-structure on $\{t\} \times M^6$
$\DeclareMathOperator\SU{SU}$I am reading Jason Lotay and Goncalo Oliveira's paper $\SU(2)^2$ invariant $G_2$-instantons, and have a few questions from the same.
If we consider the space $M = S^3 \...
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Computing/estimating geodesics in practice
Let's say I have a Riemannian manifold with known metric $g$. I want to compute the geodesics of this manifold, specifically with respect to the Levi-Civita connection.
In practice, (i.e. with a ...
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Bounded and Lipschitz De Rham cohomology representatives for pull-backs from classifying spaces
Let $(M,g)$ be a Riemannian manifold with $\pi_1(M)=\Gamma$. Let $\tilde{M}$ be its universal cover, and let $f\colon M\to B\Gamma$ be a classifying map.
Given any smooth differential form $\omega$ on ...
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implicit function theorem on manifold
Suppose that $M\subseteq \mathbb R^D$ is $d$-dimensional compact submanifold with $0\in M$, having reach $\tau>0$. Thus, for every $p\in M$, $\exp_p:B_{T_pM}(p,\tau)\rightarrow B_M(p,\tau)$ is a ...
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Differential equation on a Riemannian manifold
Let $(M,\mu)$ be a compact Riemannian manifold of dimension $n$ and $\theta$ a differential 1-form. Write $\theta=dg+\delta\phi+h$, the decomposition of $\theta$ according to the Hodge decomposition. ...
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Constructing a new manifold with a germ of manifold [closed]
Given a germ of manifolds and compatible Riemannian metrics, can we construct a new Hausdorff manifold using the exponential map?
A germ of manifolds at a point $m$ is a series of manifolds $U_i$ ...
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Non-trivial $\mathbb{R^3}\rightarrow\mathbb{R^3}$ maps with constant singular values
It can be proved that all $\mathbb{R^2}\rightarrow\mathbb{R^2}$ mappings with constant singular values are affine. In three dimensions, however, there are non-trivial examples, like
$$
\begin{align}
...
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Equivalence of Sobolev spaces for different metrics
Consider $M$ a manifold and $g_1, g_2$ two different Riemannian metrics. I want to know how the condition $|\nabla^{g_1,k}(g_1-g_2)|_{g_1}\leq C$ implies that the norms of $|\nabla^{g_1,i}u|_{T^{\...
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An integration identity on $\mathbb{P}^{n-1}$
Let $\omega_{\text{FS}}$ denote the Fubini–Study metric on $\mathbb{P}^{n-1}$ with unit volume, and let $[w_1 : \cdots : w_n]$ be standard unitary homogeneous coordinates. On page 5 of Yang–Zheng's ...
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Local variation of normals on Riemannian submanifolds of $\mathbb R^D$
Let $M^d\subseteq\mathbb R^D$ be a compact manifold, $p\in M$ and $\nu\in N_p(M)$. Suppose that all the principal curvatures in direction $\nu$ are in $[a,b]$ for some $0<a<b<\infty$. Suppose ...
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Existence non-trivial parallel $p$-form implies non-triviality of $p$-th cohomology group using De Rham cohomology
Cross-post from MSE.
Suppose $(M,g)$ be a closed Riemannian manifold. Because every parallel (nontrivial) $p$-form $\omega$ is harmonic so the $p$-th Betti number should be positive i.e. $b_p\geq 1$. ...
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Is there an area-preserving concentric diffeomorphism of the ellipse?
$\DeclareMathOperator\Vol{Vol}$This is a cross-post.
Let $0<b<1$ be a fixed parameter, and let $(R(\theta),\theta)$ be the polar coordinates of the ellipse
$$E=\{(x,y) \in \mathbb R^2 \, | \, \...
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Upper bound on index of geodesics in terms of length
Let $(M, g)$ be a compact Riemannian manifold. Let $i(-)$ be the index of a geodesic and let $l(-)$ be the length. Is there an inequality of the form $i(\gamma) \leq C l(\gamma)$ for some $C>0$ ...
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Rational systole of a manifold
I also posted this question on MSE, but since it may be a delicate question, I decided to post it here.
Given a Riemannian manifold $(M^n,g)$ and an integer $1 \leq k \leq n-1$, the $k$-systole of $M$ ...
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Gradient of solution to heat equation under evolving metric
The following simple question came to me when I was studying the heat equation on a Riemannian manifold: Suppose $M$ is a closed Riemannian manifold and $g_t$ is a smooth family of Riemannian metrics ...
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Cohomological dimension bounds on the fundamental group of a manifold
Suppose $M$ is a (closed, connected, oriented, smooth) manifold.
If $M$ is aspherical, i.e., if the inversal covering $\tilde{M}$ is contractible, $M$ is a $B\pi_1(M)$. This is often enforced by ...
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Orthonormal frame on $\mathbb{S}^3$ orthogonal to foliations [duplicate]
Does there exist a smooth orthonormal frame $X_1,X_2,X_3$ on $\mathbb{S}^3$ such that the distribution spanned by $X_i$ and $X_j$ is integrable for all $1\leq i,j\leq 3$?
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Metric deformations from non-negative to positive curvature
Is it possible to deform the metric $g$ of a closed Riemannian manifold $(M,g)$ satisfying $\mathrm{Ricci}(M,g) > 0$ and $\mathrm{sec}(M,g) \geq 0$ to a metric $g_1$ satisfying $\mathrm{sec}(M,g_1) ...
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Is existence of a limit cycles an obstruction for a vector field to be a global Jacobi field?
Is there a Riemannian metric on $S^2$ and a vector field $X$ on $S^2$ with the following two properties?
The vector field $X$ is globaly a Jacobi field in the sense that for every point $x\in S^2$ ...
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Conformal changes of metric and Ricci curvature
Let $(M,g)$ be a three dimensional smooth Lorentzian manifold and let $p$ be a fixed point in $M$ and let $S$ be a smooth symmetric tensor of rank two on $T_pM\times T_pM$. Does there exist a smooth ...
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stability of two-sided sectional curvature bounds in Lorentzian geometry
Suppose that $(M,g)$ is a Lorentzian manifold of signature $(-,+,\ldots,+)$. Given a two plane $\Pi=\textrm{Span}\{X,Y\}$ with $X,Y \in T_pM$, we say that $\Pi$ is non-degenerate if
$$ g(X,X)g(Y,Y)-g(...
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Does every group arise as the fundamental group of a complete Kähler manifold?
The fundamental group of a manifold is countable, and every countable group $G$ arises as the fundamental group of a (smooth) manifold; see this comment or this answer for a construction of an open ...
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Manifolds admitting flat connections
For each Riemannian manifold one can construct the Levi-Civita connection. While this connection is unique, we can call a (Riemannian) manifold flat if the Levi-Civita connection is flat. However when ...
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Do asymptotically conformal maps converge to a weakly conformal map?
$\newcommand{\CO}{\text{CO}_2}$
$\newcommand{\SO}{\text{SO}_2}$
$\newcommand{\dist}{\text{dist}}$
$\newcommand{\M}{\mathcal{M}}$
$\newcommand{\N}{\mathcal{N}}$
Let $\M,\N$ be two-dimensional smooth, ...
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Reference request: extendability of Lipschitz maps as a synthetic notion of curvature bounds
In the lecture Notions of Scalar Curvature - IAS around 8:00, Gromov states the following result, which he claims he does "slightly uncarefully":
Suppose $(X,g_X)$ and $(Y,g_Y)$ are ...
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Does $\pi_k(M)\neq 0$ implies $\operatorname{ind}(\gamma) < k$?
Cross post from MSE. and sorry if this is an obvious question.
Here is a line of proof of Theorem 1.15 from
Brendle, Simon, Ricci flow and the sphere theorem, Graduate Studies in Mathematics 111. ...
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