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.
3,083 questions
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Normal fields of geodesic spheres
This question is related to this one (https://math.stackexchange.com/questions/1383511/normal-curvature-of-geodesic-spheres) I've asked at math.stackexchange. Let $(M,g)$ be a compact Riemannian ...
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Examples on small cut radius of totally convex set in non-negatively curved manifold
Suppose $M^n$ is an open complete nonnegatively curved Riemannian manifold. In Cheeger-Gromoll's proof of the soul theorem. They need an estimate on the cut radius of a totally convex set $C$. By a ...
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Prescribing Gaussian curvature
Let $K(r)$ be the piecewise function
&...
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Orbits of Metrics under the Action of the Diffeomorphism Group
Consider the $n$-sphere $$ S^n = \{x\in\mathbb{R}^{n+1}: 1 - \sum_{k=1}^{n+1} x_k^2 = 0\}, $$ and let $g_1$ be the induced metric. Given $\lambda\in\mathbb{R}^{n+1}_{>0}$, we have the ellipsoid
$$
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Formula for the distance in noncommutative geometry
Probably the most famous formula in noncommutative geometry is the following formula allowing one to compute distance of two points using the operator theoretic data:
$$(1) \ \ d(p,q)=\sup\{|f(p)-f(q)|...
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Limited expansion of mean curvature of geodesic spheres
I am working with the Laplacian on a Riemannian manifold $(M,g)$ (compact, without boundary). In spherical geodesic coordinates $(r, \sigma)$ around some arbitrary $x \in M$ (where $\sigma$ denotes ...
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Immersed surfaces in Hyperbolic 3-manifolds
Given a hyperbolic 3-Manifold $M=\Gamma_{0}\setminus\mathbb{H}^3$, and a smooth, connected, compact immersed negatively curved surface $\Sigma=\Gamma\setminus\widetilde\Sigma\subset M$, where $\...
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understanding geometry of eigen values of Ricci tensor [closed]
As per I can visualize the eigen value $\lambda$ of a linear map $T:V \rightarrow V$, defined by $Tv=\lambda v$, is actually the scaling factor of the vector in the same direction as of $v$.My ...
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Symmetries of non-Riemannian curvature tensor
The curvature tensor, $R_{ab}{}^c{}_d$, can be obtained from a connection which not necessarily is a metric connection.
By construction it is antisymmetric in the first two indices, since roughly ...
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Smooth manifolds for which every metric is geodesically convex
Are there non compact smooth manifolds which have the property that every Riemannian metric is geodesically convex?
Note that a manifold for which every Riemannian metric is complete must be compact.
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Action generated by geodesic flow is hamiltonian
I'm trying to understand why a certain action of a Lie Group is hamiltonian.
Let $(M,g)$ be a geodesically complete Riemannian manifold.
Then there exists a canonical one-form on the cotangentbundle $...
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Triangles in rigid Riemann surfaces
Edit: We thank Vladimir Matveev for his comment on this post which leeds us to revise the question as follows:
Assume that $M_{g}$ is a compact Riemann surface with constant negative cuvature (That ...
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Gromov's compactness theorem for manifolds with boundary
The Gromov's compactness theorem says that if $\{M_i^n\}$ is a sequence of closed Riemannian manifolds of dimension $n$ with uniformly bounded diameter and uniformly bounded from below Ricci curvature ...
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Metrics on the 3-sphere with knotted geodesics
According to answers to this question every metrics on $S^3$ admits a simple closed geodesic. Given a knot (or link) $K$, it's also quite simple to build a metric on $S^3$ such that $K$ is a geodesic (...
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Barycentric interpolation in hyperbolic triangles
Let $T$ and $T'$ be triangles in the hyperbolic plane $\mathbb{H}^2$, denote by $A, B, C$ and$A', B', C'$ their vertices respectively. Let $f : T \to T'$ be the unique "barycentric interpolation" that ...
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$Pin^{+}(4k)$ and $Pin^{-}(4k)$ are isomorphic [Reference Request]
This is some sort of "follow-up" to the (unanswered) question posted here.
Let's denote $$\varphi :O(2n)\rightarrow O(2n);A\mapsto det(A)\cdot A.$$
Then $\varphi $ is an automorphism of $O(2n)$, and ...
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Non-flat totally geodesic surfaces
I'd like to know whether a Riemannian symmetric space of compact type admits a non-flat totally geodesic surface. I've found an article by Mashimo on the classification of these surfaces for certain ...
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Determinant twist and $Pin _{\pm}$ structure on $4k$-dimensional bundles [Reference request]
Consider the automorphism $\varphi$ of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$. The induced map in cohomology $H^*(BO(2n))$ sends $w_2$ to $w_2+(n+1)w_1^2$ (a proof is given at the end of the ...
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What dimension bound is known on the singular set of a linear combination of eigenfunctions of Laplacian?
Let $(M,g)$ be a smooth, closed Riemannian manifold and suppose that $\phi_1,\dots,\phi_m$ are eigenfunctions of the Laplacian on $M$. Write $f = \phi_1 + \dots + \phi_m$.
How big can the set $\...
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Sobolev chain rule on non-compact manifolds
Let $(M,g)$ be a non-compact Riemannian manifold (not of bounded geometry).
Let $f:\mathbb{R} \to \mathbb{R}$ be $C^1$ with $f'$ bounded and $f(0)=0$. Is the Sobolev chain rule valid for functions $...
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Difference between parallel transport composed with exponential maps along two different geodesics starting at the same point?
I asked this question on math.stackexchange too: it's not a homework problem, but something that came to my mind while thinking of commutation:
https://math.stackexchange.com/questions/1356518/...
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Flat manifolds and irreducible representations
Let $M$ be a compact Riemannian manifold with vanishing curvature of Levi-Civita connection. Such manifolds were classified by Bieberbach; sometimes they are called Bieberbach manifolds. According to ...
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Prove that the Log-Euclidean distance is negative-definite
Let $\Bbb{S}_{++}^n$ be the $\frac{n(n+1)}{2}$-dimensional Riemannian manifold of the symmetric positive definite (SPD) $n\times n$ real matrices.
The Log-Euclidean distance between two points of $\...
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Volume of arithmetic quotients of symmetric spaces
Now let $\textbf{G}$ be some connected semisimple linear algebraic group over a number field $F$. Let $G_{\infty}$ be $\textbf{G}(\mathbb{R}\otimes_{\mathbb{Q}} F)$. Let $K_{\infty}$ be a maximal ...
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Does null geodesic flow live on a natural compact bundle?
Let $(M,g)$ be a compact pseudo-Riemannian manifold (closed or with boundary).
A geodesic $\gamma:(a,b)\to M$ is called null if $g_{ij}\dot\gamma^i\dot\gamma^j=0$.
The geodesic flow can be seen as a ...
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Iwaniec's conjecture
Does anyone know whether there is any geometric applications of the Iwaniec's conjecture on $ l^p $ bound of Beurling Alfhors transform (or the complex Hilbert transform). One application could have ...
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Can every hyperelliptic genus 3 surface be minimally immersed in flat $T^3$
Every minimally immersed genus 3 surface in flat $T^3$ must be hyperelliptic, as the Gauss map gives the degree 2 covering map. How about the converse of this problem?
The only thing I can find is ...
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Diffeomorphism variation of the Christoffel symbol
Under an infinitesimal diffeomorphism the Riemann metric changes by the Lie derivative
$$
\delta g_{\mu\nu} = ({\mathcal L}_\xi G)_{\mu\nu}=\nabla_\mu \xi_\nu+\nabla_\nu \xi_\mu
$$
and under a change ...
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Decomposition of a closed surface
I know that I can decompose an hyperbolic closed surface of genus $g>1$ into $2(g−1)$ pants bounded by $3$ geodesics. It seems reasonable to think the same can be done for a closed surface of genus ...
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Almost complex structure and nontrivial idempotents
Is there a compact Reiemannian manifold $M$ for which the following complex $C^{*}$ algebra does not have a nontrivial idempotent:
$A=Hom(E,E)$ where $E$ is the complexification of $TM$.
Of ...
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Bounding distance between geodesics in manifolds with nonpositive curvature
This is a duplicate of a question at the stackexchange which was not answered. I've recently read (in some notes by Mark Pollicott) the following related claims, which, although quite intuitive, I ...
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Taylor expansions of Riemannian exponential map and Jacobi fields? [closed]
Apologies if this is not exactly a research-level questions, but I've no known reference where I can figure it out myself. I asked this on math.stackexchange.com,
https://math.stackexchange.com/...
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Involution on the set of isomorphism classes of 2n-dimensional vector bundles induced by an involution of $O(2n)$
Denote by $\varphi$ the automorphism of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$.This induces a self-map $B\varphi$ of $BO(n)$, so it induces a self-map (actually an involution)
$B\varphi ^*$ on $[...
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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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Existence, uniqueness, and smoothness of a solution to a first order PDE on Riemannian M
Let $\mathcal{M}$ be an $n$-dimensional compact Riemannian manifold, and let $\mathcal{A} \subset \mathcal{M}$ be an $n$-dimensional Riemannian submanifold. I wish to determine the local existence, ...
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Is the structure constant additive on connected components?
This is the reanimation of a question which already got an answer, that I did not fully understand. Coming back to it, after let it sit in a corner for some time, I keep not getting the point. I would ...
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Geodesics on convex hypersufaces
Let $M^n$ be the boundary of a convex compact set in $\mathbb{R}^{n+1}$ with non-empty interior.
Question 1. Is $M$ geodesically complete, i.e. is it true that every geodesic (= locally shortest path)...
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Intersections of open balls in manifolds
This question is motivated by the post Uncountable intersections of open balls in a separable metric space.
The general problem is the following: given a connected Riemannian manifold $M$, what are ...
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Gauss-Bonnet formula for 2-dimensional Alexandrov spaces
EDIT: Let $S$ be a closed orientable 2-dimensional surface equipped with a metric with curvature $\geq \kappa$ in the sense of Alexandrov.
Questions 1. Can one define a measure $K$ on $S$ (thought ...
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Taylor expansion in Riemannian foliations
Take:
$M$ a Riemannian manifold, ${X_0}\in M$,
$N_{X_0}$ a submanifold of $M$ going through ${X_0}$,
and $Z \in N_{X_0}$ in a neighborhood of ${X_0}$.
At ${X_0} \in N_{X_0}$, we consider the ...
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What is the relation between two Riemannian metrics with the same Riemannian curvatures?
Consider two flat Riemannian metrics on a manifold. What is the general relation between these two metrics if the manifold is not simply-connected? What is the answer if two Riemannian metrics have ...
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What are Euler density and Weyl invariants?
I would like to know as to what is the definition and significance of what are called "Euler density" and "Weyl invariants" (of weight $-d$ on a $d-$manifold)
Do many (which?) of them vanish when ...
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Alexandrov spaces which are not limits of Riemannian manifolds
Are there important/ interesting/ natural examples of compact Alexandrov spaces with curvature bounded from below which are not Gromov-Hausdorff limits of smooth compact Riemannian manifolds with ...
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Totally geodesic submanifold of a hyperbolic 3-manifold
If $M$ is a convex-cocompact hyperbolic 3-manifold, and $S$ is a closed surface with genus $\geq$ 2. Suppose $f:S\to M$ is a minimal immersion, and $f(S)$ is negatively curved. I know that all the ...
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Using Jacobi fields to approximate parallel transport along geodesic:is the following limit true?
I apologize if this is not a research level question (already tried asking https://math.stackexchange.com/questions/1303288/relation-between-parallel-transport-and-jacobi-field-iion stack exchange ...
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Lower bound on the first eigenvalue of the Laplace-Beltrami on a closed Riemannian manifold
It has been proved by Li-Yau and Zhong-Yang that if $M$ is a closed Riemannian manifold of dimension $n$ with nonnegative Ricci curvature, then the first nonzero eigenvalue $\lambda_1(M)$ of the (...
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A surface on which all regular curves have nowhere vanishing curvature
Let $S$ be a surface in $\mathbb{R}^{3}$ such that every regular curve $\gamma\subset S$ has nowhere vanishing curvature, that is $\kappa(z)\neq 0$ for all $z\in \gamma$. Does this imply that $...
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Spherical cap is the only compact constant mean curvature surface bounded by a circle
I would like to see that the only compact rotationally invariant constant mean curvature surfaces with boundary a planar circle, are either a planar disk or a spherical cap.
This is stated in the ...
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Riemann isometry vs Euclidean bi-Lipschitz mapping
Assume that $\gamma$ is a rectifiable Jordan curve in the complex plane of length $2\pi$. Then there exists a Riemann isometry $f$ between $\gamma$ and the unit circle $T$. My question is, does this ...
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Can a cylinder be regarded as a Riemannian manifold? [closed]
Consider the surface of a bounded cylinder consisting of a top,bottom and side part together with the metric induced by the euclidean norm on $\mathbb{R}^3$. Can this space be regarded as a Riemannian ...