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.
1,045 questions with no upvoted or accepted answers
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Minimal volume of 4-manifolds
This question came up in a talk of Dieter Kotschick yesterday. The minimal volume of a manifold is the infimum of volumes of Riemannian metrics on the manifold with sectional curvatures bounded in ...
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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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Characterising critical points of $E(f)=\int_{M}| \bigwedge^2 df|^2 \text{Vol}_{M}$
$\newcommand{\id}{\operatorname{Id}}$
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$\newcommand{\Hom}{\operatorname{Hom}}$
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Are the eigenvalues of the Laplacian of a generic Kähler metric simple?
It is a theorem of Uhlenbeck that for a generic Riemannian metric, the Laplacian acting on functions has simple eigenvalues, i.e., all the eigenspaces are 1-dimensional. (Here "generic" means the set ...
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Schoen and Yau's proof of the higher dimensional positive mass theorem
In April 2017 Schoen and Yau posted on the arxiv their solution of the time-symmetric positive mass theorem in all dimensions, which has been a significant conjecture since the 70s. As of now, July ...
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Homeomorphisms of the sphere mapping a geodesic triangulation to another one
Consider the standard Riemannian 2-sphere $S$, equipped with a geodesic triangulation $T$. Let $L(S,T)$ be the space of homeomorphisms of $S$ which map
$T$ to a geodesic triangulation. What is the ...
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Negative Einstein manifolds
In Besse's "EInstein manifolds", p. 354, the question is posed if the volume of Einstein metrics on a given compact manifold (normalized such that $Ric=\pm(n-1)g$) take only finitely many values.
For ...
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Is the oriented bordism ring generated by homogeneous spaces?
I am trying to find a Riemannian geometrically well-understood set of generators of the oriented bordism ring, including the torsion parts. By a set of generators, I mean that the set generates the ...
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Applications of Berger's Curvature Estimate
I'm interested in applications of the following estimate of Berger on the Riemann curvature tensor:
Let $(M,g)$ be a Riemannian manifold of dimension $n \geq 4$, let $p \in M$, and assume that the ...
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Are harmonic mappings non-singular outside a set of measure zero?
Let $g$ be a smooth Riemannian metric on the closed $n$-dimensional unit disk $\mathbb D^n$.
Let $f: \mathbb D^n \to \mathbb{R}^n$ be a smooth orientation-preserving immersion, and let $\omega :\...
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Best metrics on exotic R^4
What is known about the existence of complete metrics with good properties (e.g., Einstein, constant scalar curvature, etc...) on exotic ${\bf R}^4$s? Note, that some exotic ${\bf R}^4$s have non-...
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dual to Hodge theory
Let $(M,g)$ be a closed Riemannian manifold.
In my understanding Hodge theory shows that any de Rham cohomology class can be represented uniquely by a harmonic form. Moreover the harmonic form ...
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Are there $n$ points dividing a compact Riemannian manifold into equal regions?
Let $M$ be a compact, connected $m$-dimensional Riemannian manifold, and let $n\in\mathbb{N}$. Can we always find distinct points $p_1,\dotsc,p_n\in M$ such that for $i=1,\dotsc,n$ the regions $A_i=\{...
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Symmetric spaces are quandles. Is this important?
For concreteness, let's consider a connected reductive Lie group $G$, and an involution $\theta$ on it. Then the associated symmetric space $X=G/G^\theta$ has the structure of an involutive quandle: ...
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Is there any connection between the Deturck trick and the Uhlenbeck trick?
There are two separate places where ingenious uses of gauge transformations simplify the analysis of Ricci flow considerably.
The Deturck trick is a way to break the diffeomorphism invariance of the ...
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Jacobi fields on non-geodesic curves
The point of Jacobi fields is to study variations of geodesics through geodesics, but the Jacobi equation $D_t^2 J + R(J,\dot\gamma)\dot\gamma=0$ makes sense for any curve $\gamma$, not just for ...
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Elementary-ish geometric proof of Hirzebruch signature theorem for Riemannian 4-manifolds?
The Hirzebruch signature theorem tells us that for a smooth compact oriented 4-manifold, the signature $\sigma(M)$ is proportional to the first Pontryagin number of $M$:
$$
3\sigma(M)= p_1(M) = k \...
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Two ways a manifold can have little symmetry
Let $M$ be a closed connected smooth oriented manifold. The following two properties - that $M$ can either enjoy or not - intuitively both mean that $M$ has very little symmetry:
(a) Every self-map $...
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A variation on the local Günther inequality
This question is about a variation on the Günther (also known as Günther-Bishop) inequality for manifolds of sectional curvature bounded from above. With Greg Kuperberg, we would deduce from it a ...
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Elliptic regularity of perturbed scalar curvature in Kazdan & Warner
In their paper A Direct Approach to the Determination of Gaussian
and Scalar Curvature Functions, Kazdan and Warner claim something along the lines of: if $g$ is a metric in $W^{2,p}$ ($p>n$) whose ...
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Spanning curves by flat surfaces
Given a smooth closed connected curve $\gamma$ in $\mathbb R^3$, is there an immersed surface $S$ with boundary, such that its Gaussian curvature is equal to zero and $\partial S=\gamma$?
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Why are conformal transformations so relevant?
I have been studying construction of initial data in general relativity for many years now and it turns out that the most efficient methods to construct such data rely at some point on conformally ...
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Closed geodesics on closed manifolds
There is (or was) a notorious open problem (see e.g. this survey by Keith Burns and Vladimir Matveyev from 2013) in differential geometry:
Conjecture. Let $M$ be a closed (compact, with empty boundary)...
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Metrization of projective manifolds
A modern take on Hilbert's fourth problem could be as follows:
Given a manifold $M$ with a flat projective structure (i.e., a $(PGL(n+1),\mathbb{RP}^n)$-structure), find all metrics for which the ...
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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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Reference for sets of locally finite perimeter on Riemannian manifolds
I am looking for a reasonably complete reference for Ennio De Giorgi's theory of sets of locally finite perimeter (also christened by him as Caccioppoli sets, after Renato Caccioppoli's pioneering ...
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k-th Pontryagin class of $\Lambda^{2k}_{\pm}$ on an oriented $4k$-manifold
If $M^{4k}$ is an oriented Riemannian $4k$-manifold, then the star-operator splits the bundle $\Lambda^{2k}$ into $\pm 1$-eigenspace bundles denoted $\Lambda^{2k}_{\pm}$.
I'm curious if anyone has ...
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Local meaning of the Pfaffian of the curvature
The Ricci and scalar curvatures have very nice pointwise interpretations (using the local expression for the volume form for example). So, (at least Ricci) having special metrics (like Einstein) can ...
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Comparing spectra of Laplacian and Schrödinger operator
Let $M$ be a closed (compact without boundary) Riemannian manifold. Is there a body of results that compares the eigenvalues of the Laplace-Beltrami operator with that of Schrödinger operators $-\...
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Is it overkill to invoke Kirszbraun theorem to prove the following fact ?
Given a small enough convex triangle $(abc)$ in a (smooth or Alexandrov) surface $(X,d)$ of curvature greater than $-1$, let $(\overline{abc})$ be its comparison triangle in $\mathbb{H}^2$. Then there ...
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Killing spinors and symmetric tensor fields.
Hi all,
I have a question of the following form: Let $(M,g)$ be a Riemannian spin manifold which admits a Killing spinor $\sigma$ and let $h:T M \to T M$ be a symmetric, trace-free and divergence-...
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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 $$\mathrm{...
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Explicit construction of a (the?) dual symmetric space
I am looking for a reference, proof or disproof of the fact that every Riemannian globally symmetric space of compact (non-compact) type has a "dual", which is of non-compact (compact) type.
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Perturbing metrics with nonpositive curvature
Let $M$ be a compact $3$-dimensional manifold diffeomorphic to a ball. Suppose that $M$ has nonpositive (sectional) curvature and its boundary $\partial M$ is convex, or even that $M$ is a Riemannian ...
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Is there a variational interpretation for the equation $\operatorname{div}(\star \circ \bigwedge^k df\circ \star )=0$?
$\newcommand{\id}{\operatorname{Id}}
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approximation of currents
Let $M$ be a closed Riemannian manifold of dimension $d$. Let $d \alpha$ be a smooth exact $p$-form. We define a current $T_{d \alpha}$ as follows : for any smooth $(d-p)$-form $\beta$ we set
$$ T_{d \...
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Nash embedding for 3 manifolds
The Nash embedding theorem tells us that every smooth Riemannian m-manifold can be embedded in $R^n$ for, say, $n = m^2 + 5m + 3$ (edit: 14 is a better bound for compact 3 manifolds thanks @mme). What ...
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Volume of a geodesic ball in $\operatorname{SL}(n) / {\operatorname{SO}(n)}$?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}$Crossposted on MSE: https://math.stackexchange.com/questions/4261809/volume-of-a-geodesic-ball-in-sln-son
Question: What is the volume of a ...
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Maximal geodesic spheres in the "octooctonic projective plane"
Boris Rosenfeld claimed that the 128-dimensional compact Riemannian symmetric space on which $\mathrm{E}_8$ acts as isometries could be seen as the "octooctonionic projective plane", $(\...
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Hermitian sectional curvature
Let $N$ be a Riemannian manifold, denote $R$ its purely covariant Riemann curvature tensor with sign convention so that the sectional curvature is $K(X,Y) = R(X,Y,X,Y)$ for an orthonormal pair.
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Diffeomorphism type of Ricci-flat four manifolds
Let $(M,g)$ be an irreducible compact and simply connected Ricci-flat Riemannian four-manifold. My first questions are as follows:
A) Is there a classification of the possible homeomorphism types of ...
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Counter-examples to the higher dimensional statement of the half-space theorem
The well-known Half-space Theorem by Hoffman and Meeks says that there is no nonflat complete properly embedded minimal surface contained in an half space of $\mathbb{R}^3$.
The higher dimensional ...
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Existence of barycenter
Let $(X,d)$ be a metric space. A barycenter of a Borel probability measure $\mu$ on $X$ is a minimizer of the function
\begin{equation}
\begin{split}
f \colon X & \to \mathbb{R}\\
x &\mapsto \...
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Second Stiefel-Whitney class as an obstruction to the existence of spin structure
Let $M$ be an oriented (closed) Riemannian manifold. Choose a good open cover and local trivialisations of the tangent bundle $U_i$. Then we get a system of transition functions $\varphi_{ij}: U_i \...
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Have heat kernels for generalized Laplacians on non-compact manifolds been constructed?
Let $M$ be a non-compact Riemannian manifold which is "nice enough", and $D$ a generalized Laplacian on it. The construction of the heat kernel for the Laplace-Beltrami operator on $M$ seems to be ...
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Tangent space, metrics etc. on simplicial sets
Is there a way to attach some sensible notion of tangent space to a simplicial set? If yes, is it possible to transfer typical local data from differential geometry such as metrics to this setting?
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Sobolev embedding theorems in vector bundles on non-compact manifolds
Let $(M,g)$ be a smooth (not necessarily compact) Riemannian $n$-manifold. It is well-known that dealing with Sobolev spaces in the general non-compact case becomes tricky, since for instance, there ...
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What specifically is the gap in Aubin's argument about positive Ricci curvature that Paul Ehrlich alludes to?
In his paper [2], Paul Ehrlich write
In [1], Aubin stated a theorem which implied as a corollary that if a manifold
$M$ admits a Riemannian metric with nonnegative Ricci curvature and
all Ricci ...
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Intuition for the volume form - combinatorial definition?
I apologize that this is short of research level but I have realized that I am not happy with my understanding of the volume form on an oriented Riemannian manifold and I was hoping to find some ...
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