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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Complete stable minimal hypersurface in positively curved manifolds
Let $(M^n,g)$ be a complete noncompact orientable Riemannian manifold with positive sectional curvature. Can we find an orientable stable minimal hypersurface $N$ in $M$?
It follows from R. Schoen's ...
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Asymptotic expansion of the Schrödinger kernel?
My stackexchange post was somewhat unsatisfactory (also because I may not have stated clear enough what my interest was). So here it goes!
Let $M$ be a compact Riemannian manifold and $\Delta$ be the ...
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Why 2-tori with Gauss curvature $\geq -1$ cannot collapse to segment?
Let $\{(\mathbb{T}^2,g_i)\}_{i=1}^\infty$ be a sequence of 2-dimensional tori with smooth Riemannian metrics with Gauss curvature at least $-1$. It was explained in the final answer to the post Gromov-...
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The semigroup of Laplace-Beltrami operator on 3-flat torus
I am studying a recent paper in which the author worked on the rectangular, flat 3 tori. It can be realized, the author explained, as $\mathbb{R}^3 \over (L_1 \mathbb Z \times L_2 \mathbb Z \times L_3 ...
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PDE system solution on manifold with conformal metric
Let $(M, \bar g)$, where $M=\mathbb{R}^n$, with coordinates $(x_1, x_2, .. x_n)$, $f:\mathbb{R}^n \rightarrow \mathbb{R}$ and $\phi:\mathbb{R}^n \rightarrow \mathbb{R}$ with $n \geq 3$ and $f$ a ...
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Distance and initial velocity of the shortest path along a smooth curve in a manifold
Let $(M,g)$ be a Riemannain manifold and let $p\in M$. Let $\gamma:[0,1] \to M$ be a smooth curve and let $p \notin \gamma([0,1])$. Assume further that for each $t \in [0,1]$ there is a unique (unit ...
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Geodesic foliations of open manifolds foliated by hyperbolic spaces
It is known that hyperbolic spaces admit geodesic foliations (that is, a smooth unit vector field all of whose integral curves are geodesics, see https://arxiv.org/abs/1411.6700).
Suppose a complete ...
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Does every non-compact hyperbolic manifold admit compact complex submanifolds?
Let $(X,\omega)$ be a complete Kähler manifold with a metric of negative holomorphic sectional curvature. Does $X$ admit a proper, positive-dimensional, compact complex submanifold?
In general, it is ...
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Exponential decay of resolvent kernel
For the integral kernel of the Laplacian $\Delta$ on $\mathbb{R}^n$, consider the resolvent $R(\lambda) := (\lambda - \Delta)^{-1}$ and let $R(\lambda; x, y)$ be its kernel, which is a smooth function ...
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Weyl asymptotic in towers
Let $M$ be a compact Riemannian manifold. The Weyl law gives the asymptotic of the counting function $N(T)$ of Laplace eigenvalues $|\lambda|\le T$ as $T\to\infty$.
Now suppose you are given a tower ...
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Relation between Ricci curvature and sectional curvature for 3-manifolds
Let $(M^n,g)$ be a smooth Riemannian manifold. It is well known that if $sec(M)\geq \kappa$ then $Ric(M)\geq (n-1)\kappa$.
If I understand correctly in dimensions $n\geq 4$ a lower bound on $Ric(M)$ ...
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Possible sign of scalar curvature for Einstein warped product manifold with Ricci-flat
Let $(M, g_M)$ where $M= B \times_f F$ and $g_M=g_B + f^2g_F$, an Einstein warped product manifold (i.e., $Ric_M= \lambda g_M$), with Ricci flat fiber-manifold $F$, i.e., $Ric_F=0$.
Then $M$ can admit ...
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Is every homogeneous space Riemannian homogeneous?
A manifold $M$ together with a transitive $G$-action is always diffeomorphic a quotient $G/H$ for $H < G$ Lie groups. On the other hand, there might be a proper subgroup of $G$ that also acts ...
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geometry and connected sum of aspherical closed manifolds
Let $ G $ be a Lie group with finitely many connected components, $ K $ a maximal compact subgroup, and $ \Gamma $ a torsion free cocompact lattice. Then
$$
\Gamma \backslash G/K
$$
is an aspherical ...
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Proving an eigenvalue bound without resorting to Weyl's law
Suppose $(M,g)$ is a smooth compact Riemannian manifold of dimension $n\geq 2$ with smooth boundary and denote by $\{\phi_k,\lambda_k\}_{k\in \mathbb N}$ its Dirihclet spectral decomposition for the ...
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Weyl's law and eigenfunction bounds for weighted Laplace-Beltrami operator
I would appreciate any answers or even references for the following problem.
Let $(M,g)$ be a complete smooth Riemannian manifold with an asymptotically Euclidean metric (let's even say that the ...
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Global choice of eigenvectors on an open surface
Let $(M^2,g)$ be a noncompact orientable Riemannian surface without boundary. Let $A \in \Gamma(\operatorname{Sym}(TM))$ be a section of the bundle of symmetric endomorphisms of $TM$, that is, for ...
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On a closed manifold whose curvature is close to "hyperbolic"
$\DeclareMathOperator\Vol{Vol}$In 1.7 on p.224 of the following paper, there is a rigidity result for compact manifolds whose sectional curvature is almost $-1$.
Gromov, M.. Manifolds of negative ...
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Compact flat orientable 3 manifolds and mapping tori
There are 10 compact flat 3 manifolds up to diffeomorphism, 6 orientable and 4 non orientable. I am looking to better understand how to construct the orientable ones.
The six orientable ones are ...
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Approximation of continuous projections on a manifold by smooth idempotents
Every continuous vector bundle on a closed smooth manifold $M$ has a smooth structure. On the other hand, every vector bundle $E$ is the image of a trivial bundle $M\times\mathbb{C}^n$ under some ...
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On Sobolev spaces on domains in Riemannian manifolds
There is extensive literature on Sobolev spaces on complete Riemannian manifolds but are there any standard references regarding the definition and properties of Sobolev spaces on domains (possessing ...
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What is the Jacobi tensor?
On the Woldram Mathworld site, tgs Jacobi tensor is defined to be:
$J^m_{ijk} :=1/2( R^m_{ijk} + R^m_{kji})$
where $R^m_{ijk}$ is the Riemann curvature tensor.
I've not been able to find anything else ...
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Is this generalization of differentiable manifolds to mixed dimensions a known object?
Suppose you are studying the evolution of some electromagnetic quantity in a conductor consisting of objects of several dimensions, i.e. wires, plates and balls.
This would amount to studying the ...
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Rademacher type theorem for Alexandrov spaces
The classical Rademacher theorem says that any Lipschitz function on a doman in $\mathbb{R}^n$ has the first derivative almost everywhere.
I am wondering if this result can be generalized as follows. ...
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Illuminating piecewise-flat manifolds with geodesics
If one issues a geodesic in every direction from a point $p$ on a piecewise-flat 2-manifold, will
it necessarily illuminate the entire surface? I know the answer is 'No,' but I would like to explore
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Extension of measurable function from dense subset
Let $M$ be a compact riemannian manifold equipped with a geodesic distance and let $\mathcal{B}(M)$ be the borel sigma algebra generated by the geodesic distance. Let $(\Omega,\mathcal{F},\mathbb{P})$...
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Mappings between 2-manifolds with symmetries with fixed singular values
Let $\left(\mathcal{M}^2,g_\mathcal{M};X\right)$ and $\left(\mathcal{N}^2,g_{\mathcal{N}};Y\right)$ be two smooth two-dimensional, simply connected Riemannian manifolds (with or without boundary), ...
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Is there a non geodesible vector field $P\partial_x+Q\partial_y$ which satisfies $P_xP_y+Q_xQ_y=0$
Inspired by the following two posts
Finding a 1-form adapted to a smooth flow
Does $P_xP_y+Q_xQ_y=0 \implies$ "non-existence of limit cycle" for $P\partial_x+Q\partial_y$"? (Complex ...
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Fundamental group of compact globally symmetric spaces
The fundamental group of a globally symmetric space $M$ of compact type is known (see Loos [1], Borel [2]). The result can be formulated as follows: it is isomorphic to the quotient
$$(*) \quad \pi_1(...
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Positive scalar curvature on the double of a manifold
Let $(M,g)$ be a compact Riemannian manifold with boundary and assume it has positive scalar curvature.
Question. Is it true that $DM$, the double of $M$, admits a metric of positive scalar curvature?...
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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}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\TM}{\operatorname{TM}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Cof}{\operatorname{Cof}}
\newcommand{\Det}{\...
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Lipschitz domains ambiguous definitions
I use a lot in the study of pde bounded Lipschitz domains $\Omega\subseteq\mathbb{R}^N$. However I have noticed that there are some major differences in their definitions. I will put here two of them, ...
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Intuition for no isometric embedding of round projective plane into $\mathbb{R}^4 $
This has been on MSE for over a month with four upvotes but no answers or even comments so I'm cross-posting:
According to Examples of two-dimensional Riemannian manifolds that can't be ...
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Classification of homogeneous Einstein manifolds
In Besse's "Einstein manifolds", p. 177, he states that, until that moment, no general classification of homogeneous Einstein manifolds was know, even in the compact case. More specifically, ...
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Essential simple closed curves in a torus [closed]
Definition: By a closed curve in a surface $S$ we will mean a continuous map $S^1 \to S$.
We will usually identify a closed curve with its image in $S$. A closed curve
is called essential if it is not ...
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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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Is there a mathematical book on general relativity that uses exclusively a coordinate free language even in practical computations?
I would also appreciate if it was as far from the physicists formalism as possible, no abstract indices ,etc. Also I don't consider using a basis or tetrads as coordinate free.
The idea is to use ...
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Embedding round manifolds into low dimensional spheres
Robert Bryant's answer to Isometric embedding of SO(3) into an euclidean space mentions that there is an isometric embedding of the round tetrahedral space $ SO_3/A_4 $ into the round sphere $ S^6 $.
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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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Riemannian homogeneous equivalent to linear group orbit
Let $ M $ be a smooth manifold.
Recall that a manifold $ M $ is smooth homogeneous if there exists a Lie group acting transitively on $ M $.
Recall that a manifold $ M $ is Riemannian homogeneous if ...
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noncompact Riemannian homogeneous is trivial vector bundle over compact homogeneous
Is it true that a manifold $ E $ admits a metric with respect to which the isometry group is transitive ($ E $ is Riemannian homogeneous) if and only if $ E $ is the total space of a $ K $ equivariant ...
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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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Examples of compact Kähler manifolds whose Bochner curvature tensor has constant norm?
The Bochner curvature tensor is the Kähler analog of the Weyl curvature tensor in the curvature decomposition of a Kähler, discovered by Bochner in 1949. The article on Bochner-Kähler metrics by ...
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Proving that a twisted cotangent bundle is geometrically bounded
In Cieliebak, Ginzburg and Kerman's paper Symplectic homology and periodic orbits near symplectic submanifolds, the authors claim and give a proof that a twisted cotangent bundle will be geometrically ...
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Classifying space $\text{BU}(n)$ from the differential-geometric point of view?
The classifying space of a topological group $G$ is the quotient of $EG$ (a topological space with vanishing homotopy groups) by a proper free action of $G$. The standard notation for the classifying ...
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The classification of developable surfaces: Are these statements equivalent?
This is a cross-post from MSE (https://math.stackexchange.com/q/4330772/242708).
I thought to know very well the answer to the classification problem for developable surfaces, so I sought for some ...
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Mapping torus of orientation reversing isometry of the sphere
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\O{O}\DeclareMathOperator\Iso{Iso}$
Let $ f_n $ be an orientation reversing isometry of the round ...
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Does Ricci flow depend continuously on the initial metric?
Consider a version of Ricci flow for which short time existence and uniqueness are known,
e.g. the Ricci flow on a closed manifold. Does the solution $g_t$ for small $t$ depend continuously on the ...
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When exponential map is 1-1 from vector fields to diffeomorphisms
Let $M$ be a connected and complete Riemannian manifold of positive dimension, $k$ be a positive integer, and let $\mathfrak{X}^k_c$ be the set of class $C^k$-vector fields on $M$ of compact support. ...
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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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