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.
348 questions
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Pólya's conjecture on the spectra of the Laplacians
Recently I've learned something about the spectra of the Laplacians. Given a bounded domain $\Omega \subset \mathbb{R}^n$ with $\partial \Omega$ smooth, we can consider eigenfunctions of Dirichlet ...
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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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Explicit formula for the Levi-Civita connection on a non-compact Riemannian symmetric space
Let $G/K$ be a non-compact Riemannian symmetric space, endowed with the Riemannian metric coming from the Killing form on the Lie algebra $\mathfrak{g}$ of the semi-simple Lie group $G$. Here $K$ is ...
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Are metric isometries smooth at the boundary?
Let $M,N$ be smooth Riemannian manifolds with boundary (In particular, we assume the boundaries are smooth).
Suppose we have a map $\phi:M \to N$ which satisfies the following properties:
$$(1) \, \,...
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Does closed Alexandrov space admit a bi-Lipschitz embedding into $\mathbb R^N$?
As the title says.
Let $A^n$ be an $n$-dimensional closed Alexandrov space. Does it admit a bi-Lipschitz embedding into Euclidean space $\mathbb R^N$ for sufficiently large $N$?
I know there are some ...
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Submanifolds of Lie groups with abelian normal bundle
Let $M$ be a submanifold of a symmetric space $Q$. The normal bundle $NM$ is called abelian if $\exp(N_{p}M)$ is contained in some totally geodesic and flat submanifold of $Q$ for all $p \in M$; see ...
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Does complexified isometry group act transitively on tangent bundle of compact Riemannian manifold?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\O{O}\DeclareMathOperator\Iso{Iso}$Let $ g $ be the round metric on the sphere $ S^n $. Since $ S^...
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Bisector of two points in a Riemannian manifold has measure $0$
Let $p,q\in M$, $p\neq q$, where $M$ is a Riemannian manifold. We will let the bisector of $p,q$ be $\mathcal{B}(p,q)=\{x\in M;d(p,x)=d(q,x)\}$. Does $\mathcal{B}(p,q)$ have measure $0$?
I was ...
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Finitely generated projective modules over the algebra of sections of the Clifford bundle
Consider a (pseudo-)Riemannian manifold $(M,g)$ and the corresponding Clifford bundle $Cl_g(T^*M)$. Let $R$ be the algebra of sections of $CL_g(T^*M)$, with point-wise multiplication. What are the ...
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minimal surfaces in $S^n$
Thanks to Choi-Schoen theorem, we know that the space of embedded minimal surfaces into $S^3$ of fixed genus is compact. My question are simples:
Can we remove the embeddness assumption?
Can we ...
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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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Eigenvalues of the Laplacian on surfaces with boundary
Let $\Sigma$ be a compact smooth surface with boundary. Is it true that the supremum
$$\sup \{ \lambda_1(\Sigma,g) \operatorname{Area}(\Sigma,g) : g \text{ smooth Riemannian metric on $\Sigma$} \}$$
...
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Fundamental groups of compact manifolds with non-negative Ricci curvature.
I would like to find an appropriate reference for the following statement:
Statement. Let $M$ be a compact Riemannian manifold with non-negative Ricci curvature.
Then $\pi_1(M)$ is virtually abelian.
...
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Changing coordinates so that one Riemannian metric matches another, up to second derivatives
Let $g$ and $g'$ be two $C^2$-smooth Riemannian metrics defined on neighborhoods $U$ and $U'$ of $0$ in $\mathbb R^2$, respectively. Suppose furthermore that the scalar curvature at the origin is $K$ ...
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Minimal distance spheres in complex projective spaces
My question has to do with distance spheres in $\mathbb CP^{n+1}$. I am interested in knowing what is the radius $r$ of a distance sphere $S(r)$ around a point that makes it a minimal submanifold of $\...
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The integral of torsion
I found the following * exercise (exercise *9) in page 407 of the book of do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced ...
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existence of totally geodesic hypersurfaces
Assume we are on a smooth, complete Riemannian manifold $(M,g), dim(M) \geq 3$. What are the specific geometric/topological constraints for such a manifold to admit complete, totally geodesic ...
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Volume of geodesic balls
I have two questions (somewhat related) regarding local geometry on a SMOOTH, COMPACT Riemannian manifold. I still have a hard time getting a "good" understanding of local geometry.
Question 1:
It ...
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Jacobi fields on a "bump surface"
Consider a "bump surface" which looks like the following:
Such a surface is rotationally symmetric, $C^2$-smooth, has positive curvature in the middle and negative curvature along the ring (the ...
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The contractivity of the heat semigroup in $L^p$ spaces
Let $M$ be a Riemannian manifold. By functional calculus, it is immediate to show that the heat semigroup is a contraction in $L^2(M)$. I can also show that it is a contraction in any $L^p(M)$ with $p ...
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Maximum symmetry metric on irreducible compact symmetric space
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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Totally geodesic subgroups in Lie groups
Let $G$ be a Lie group with a left invariant metric $g$.
Let $H$ be a (closed) Lie subgroup of $G$, and assume $g$ is right-$H$-invariant. (That is $d(R_h)_e:T_eG \to T_hG$ is an isometry for every $...
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Hypersurfaces and Elliptic Points
I'm reading a paper, in which we have $M^n$ an n-dimensional compact hypersurface embedded in $\mathbb{R}^{n+1}$. We take the scalar cuvature $R$ to be the elementary symmetric polynomial of degree 2 ...
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Ricci curvature of the symplectic group
Is the Ricci curvature of the compact symplectic group $Sp(n)$ bounded below by $cn$ for some constant $c > 0$ independent of $n$?
For $O(n)$ and $U(n)$ I know many references which state such a ...
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Unbounded sectional curvature implies infinite diameter?
Let $(M,g)$ be a Riemannian manifold such that for each $C>0$ there is $p\in M$ and $X,Y\in T_pM$ unitary such that $K(X,Y) > C.$ Does this imply that the diameter of $(M,g)$ is infinite?
I ...
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Volume doubling, uniform Poincaré, counterexample
The Poincaré inequality and the volume doubling property are important notions related to heat kernel estimates.
Pavel Gyrya and Laurent Saloff-Coste obtain the two sided heat kernel estimate of ...
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Poincare-like inequality on compact Riemannian manifolds
I am looking for a Poincare Inequality on balls but instead of euclidean space, I have a compact Riemannian manifold without boundary. The inequality I am looking for is the equivalent of
$$ \int_{...
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The distributional gradient of the closest isometry to the differential of a smooth map
The setting-a "linear algebra" fact:
Let $A$ be a real $n \times n$ matrix, and suppose that $\det A<0$ and that the singular values of $A$ are distinct. Then, there exist a unique matrix $Q(A) \...
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Does every smooth manifold of infinite topological type admit a complete Riemannian metric?
To elaborate a bit, I should say that the question of the existence of a complete metric is only of interest in the case of manifolds of infinite topological type; if a manifold is compact, any metric ...
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Counting limit cycles via curvature in Riemannian geometry
In this post we would like to give a possible new approach to the second part of the Hilbert 16th problem
First we give a short introduction:
A quadratic system is a polynomial vector field on ...
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Relation between harmonic vector field and harmonic 1-form
Definition 1: A unit vector field $X$ side to be harmonic if it is critical point for the following energy function
$$E(X)=\frac{1}{2}\int_M\|dX\|^2dvol_g=\frac{m}{2}vol(M,g)+\int_M\|\nabla X\|^...
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Examples of two-dimensional Riemannian manifolds that can't be isometrically embedded into $\mathbb{R}^4$
Can anyone give some examples of two-dimensional Riemannian manifolds $(M,g)$ that can't be isometrically embedded into $\mathbb{R}^4$? (Further more Globally)
What if it is smooth?
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Can a simple Riemannian metric on the disc be extended to a Zoll metric on the sphere?
Given a simple Riemannian metric $(D,g)$ on the two-disc---its geodesics have no conjugate point and the boundary of the disc is strictly convex---, is it possible to embed $(D,g)$ isometrically into ...
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Metric Connections on a Lie Group
A Lie group has three standard Cartan connections; the (-)-connection, the (0)-connection, and the (+)-connection. The (0)-connection is Levi-Civita with the associated metric the bi-invariant metric. ...
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Parallel transport as algebra isomorphism
Assume that there is an smooth structure of the matrix algebra $M_{n}(\mathbb{R})$ on fibers of the tangent bundle of a $n^2$ dimensional manifold.
Is there a Riemannian metric on $M$ such that all ...
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Geodesics for non differentiable riemannian metric
Let $M$ be a differentiable manifold of dimension $n>2$ with a Riemannian metric $g=\sum_{i,j=1}^ng_{ij}dx_idx_j$ such that in some points on $M$ its coefficients $g_{ij}$ are not differentiable (...
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When a Riemannian manifold is of Hessian Typ
When a Riemannian manifold is of Hessian Type (i.e., a Riemannian manifold which its metric is Hessian)
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Compact surface with genus$\geq 2$ with Killing field
Let M be a compact Riemannian surface of genus$\geq 2$.
Can M have a globally defined Killing field ?
Can M have a Killing field defined on M-(finite set of points)?
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Compact manifolds locally bi-Lipschitz to Euclidean space
I have a compact manifold $M$, and I am allowed to choose some Riemannian metric on it, exactly which I don't care. But I would love it if I could choose the metric $g$ such that every point has an ...
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Hyperbolic manifolds containing totally geodesic hypersurfaces which themselves contain totally geodesic hypersurfaces
I will preface this by saying that while I am familiar with the general theory of (semi)-Riemannian manifolds, I am a complete novice when it comes to the specifics of hyperbolic manifolds (I am using ...
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The "Rolle theorem" for sections of a vector bundle
1) Assume that $E\to M$ is a smooth real vector bundle and $\nabla$ is a connection. (We do not assume any metric compatibility since we do not fix a metric on $E$). Assume that ...
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Tetrad postulate: Implies or results from the metricity of the connection?
Hi,
I see that the tetrad postulate:
$\nabla_{\mu}e_{\nu}^{I}=\partial_{\mu}e_{\nu}^{I}-\Gamma_{\mu\nu}^{\rho}e_{\rho}^{I}+\omega_{\mu J}^{I}e_{\nu}^{J}=0$
Can be merely derived from writing a ...
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Dynamical obstructions for a vector field whose derivation sends an orthonormal set to a mutually Sasakian orthogonal vectors
We ask two related questions which are inspired by this MO question Does $P_xP_y+Q_xQ_y=0 \implies$ "NONEXISTENCE OF LIMIT CYCLE for $P\partial_x+Q\partial_y$"? (Complex Dilatation and Limit ...
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Extending Gromov's inequality
In 1981 Gromov proved that all Riemannian metrics on the complex projective space $\mathbb CP^n$ satisfy the bound
$$\DeclareMathOperator{stsys}{stsys} \DeclareMathOperator{vol}{vol}
\frac{\stsys_2^n}{...
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Manifolds with boundary admitting no closed embedded minimal hypersurface
The following Theorem is proved in the paper entitled "Compactness of the space of embedded minimal surfaces with free boundary in three-manifolds with nonnegative Ricci curvature and convex ...
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"The famous Lusternik-Schnirelmann Theorem of the Three Closed Geodesics"
The title is a quote from p.256 of Wilhelm Klingenberg's 1995
Riemannian Geometry (Google Books link):
Every surface homeomorphic to a sphere $\mathbb{S}^2$ has three distinct, simple, closed ...
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Isometric embedding of a genus g surface
Can a genus $g$ surface with constant negative curvature and $g>1$ be isometrically embedded in $\mathbb{R}^4?$
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Uniformization for annuli with boundary
Let $(A,g)$ be a compact surface with boundary, diffeomorphic to the standard annulus $\{z\in\mathbb{C}:1\le|z|\le 2\}$, equipped with a smooth metric $g$.
Does there always exist a conformal (...
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Is polar decomposition of a smooth map Sobolev?
Motivation:
Let $\mathbb{D}^2$ be the closed unit disk. I am studying the "elastic energy" functional $E(f)=\int_{\mathbb{D}^2} \text{dist}^2(df,\text{SO}_2)$, where $f \in C^{\infty}(\mathbb{D}^2,\...
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Differential equation involving Casimir operator on $\operatorname{SL}(2,\mathbb{R})$
Throughout this post, we let $G$ denote the Lie group $\mathrm{SL}(2,\mathbb{R})$. For $t,\theta \in \mathbb{R}$ we define the following elements of $G$:
\begin{align*} A(t) &= \begin{bmatrix}e^t &...
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