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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Morrey & Grauert - real analytic vector bundles admits analytic Riemannian metric
In theorem 1.2 of Brian Conrad's handout Operations with Pseudo-Riemannian metrics, the author writes
Theorem 1.2. Every $C^p$ vector bundle $E\to M$ over a $C^p$ manifold with corners $0\leq p\leq \...
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Integrability of distributions which are invariant under the isometry group
Let $(M.g)$ be a Riemannian manifold and $D$ is a distribution on $M$. Assume that $D$ is invariant under the action of the isometry group of $M$.
Under which conditions such $D$ ...
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Does convex set in Alexandrov space has positive reach?
Let $M$ be a metric space, $A$ a subset of $M$. The reach (defined by Federer) of $A$ in $M$ is the largest $r_0\ge 0$ such that if $x\in M$ and the $d(x, A)< r_0$, then $A$ contains a unique point ...
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Is every connected complete Riemannian manifold of non-positive sectional curvature a Polish space?
Question: is it true that every finite-dimensional connected complete Riemannian manifold of non-positive sectional curvature is a Polish space, i.e. homeomorphic to a complete separable metric space?
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minimal diameter of sphere with sectional curvature bounded above
Assuming that $g$ is a metric on the n-dimensional sphere $S^n(n\geq 3)$ satisfying sectional curvature bounded: $|K(g)|\leq 1$, is the diameter of $g$ at least $\pi$? How about if we only assume that ...
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Reference: Varadhan's lemma for Finsler Geometry?
Is there a version of Varadhan's lemma for heat-kernels on Finsler manifolds? I expect this to exist but I cannot seem to find any papers on the topic. References would be greatly appreciated.
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Were 3-manifolds with $\sec>0$ known to be space forms before Ricci flow?
It is well known that R. Hamilton (JDG 1982) used Ricci flow to show that a closed $3$-manifold with positive Ricci curvature must be diffeomorphic to a spherical space form $S^3/\Gamma$, since such ...
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Curvature blow up along Ricci flow
In the book on Ricci flow by Andrews and Hopper, it has been proved that if Ricci flow on $M$ has a finite time singularity at time $T$ then $\lim_{t \nearrow T} \sup_{x\in M} |Rm(x,t)|=\infty$. I am ...
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A differentiable isometry is smooth?
I posted this question in MSE but got no response (even after giving a bounty), so I am trying here.
Let $M,N$ be smooth $d$-dimensional Riemannian manifolds.
Suppose $f:M \to N$ is a differentiable ...
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For what potentials is the heat operator with a potential term hypoelliptic?
If $(M,g)$ is a Riemannian manifold and $\Delta$ is the Laplace-Beltrami (negatively defined) operator, is it possible to describe the class of smooth potentials $V :M \to \Bbb R$ that make the heat ...
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When is a six-dimensional manifold the twistor space of a four-dimensional manifold?
What are the conditions on a six-dimensional manifold to be the twistor space of a four-dimensional one?
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Surfaces in isothermal coordinates and particular PDE
From Brioschi's formula, if we have a surface in isothermal coordinates were $g_{ij}=E(u,v)*\delta_{ij}$ is the metric tensor, the gaussian curvature is:
$K=-\frac{1}{2E}[\frac{\partial}{\partial u}(...
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What are the scalar conformal invariants of weight -3/2 in 3 dimensions?
I am looking for all the scalar conformal invariants (diffeomorphism-invariant polynomials $P[g]$ in the metric $g_{ij}$, its inverse $g^{ij}$ and its derivatives $g_{ij,klm\dots}$ such that $P[\...
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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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How large can the cone of $\nabla$-compatible metrics be?
Let $E$ be a smooth vector bundle over a manifold $M$, equipped with a connection $\nabla$.
The set of $\nabla$-compatible metrics on $E$ forms a convex cone.
This cone can be empty, however (see ...
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Tweetable way to see Riemannian isometries are harmonic?
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Smooth Riemannian isometries are harmonic. Can one conclude ...
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A model of smooth projective plane
Is there a model of smooth projective plane $(P,L,F)$ with the following property?
The point space $P$ admits a Riemanian metric such that for every point $p\in P$ and every line $\ell_{1}$ ...
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Volume of $SO(n)\subset\mathbb R^{n^2}$, again
I posted this on MSE, but no answer is received, so I post this here.
The problem of finding the $n(n-1)/2$-dimensional volume of the set $SO(n)\subset\mathbb R^{n^2}$ is asked before in this MO post....
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On the isometry group of a self cartesian product of a Riemannian space
Let $X$ be a complete Riemannian space. Let us denote by $Iso(X)$ the group of isometries of $X$. It is a well-known fact that the group $Iso(X)$, when endowed with the compact-open topology, is a Lie ...
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Are all the mappings which satisfy this equation scaled isometries?
Let $M,N$ be smooth oriented $d$-dimensional Riemannian manifolds, $\, f:M \to N$ a smooth map. Let $\Omega^1(M,f^*TN)=\Gamma(T^*M \otimes f^*TN)$ be the space of $f^*TN$-valued one-forms.
Let $d$ ...
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Conformal Transformations that are Ricci Positive Invariant
Is there any known class of conformal transformations $\phi : M \to M$ of a riemannian/semi-riemanian manifold $(M,g)$ that have the property: $g$ is ricci-positive iff $\phi^* g$ ricci positive?
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Geodesics on Homogeneous Spaces of $SU(n)$
Consider the homogeneous space $SU(n)/K$, where $K$ is a sub-group of $SU(n)$ and the bi-invariant metric on $SU(n)$.
What is the appropriate quotient metric on the homogeneous space and what are the ...
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Totally convex, convex and locally convex sets
Consider the following definitions :
$C\subset M$ is convex if any $p,\ q\in C$ all minimizing
geodesic between $p$ and $q$ are in $C$
$C$ is totally convex if for $p,\ q\in C$, every geodesic ...
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G-spaces and manifolds
In his book "The geometry of geodesics" H. Busemann defines the notion of a G-space to be a space which satisfies the following axioms:
The space is metric
The space is finitely compact, i.e., a ...
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Determining the rate of spread of geodesics when the sectional curvature is zero
I have posted this question in mathSE a few weeks ago (and proposed a bounty) but so far got no response.
In the book Riemanian geometry (by do-Carmo), the following result is proved (Corollary 2.9 ...
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Brownian motion on a $\mathbb{Z}$-cover
Let $(M,g)$ a smooth closed Riemannian manifold with non trivial first homology group $H^1(M,\mathbb{R})$. Any element of $H^1(M,\mathbb{Z})$ will define a riemannian $\mathbb{Z}$-cover of $M$ ...
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Codifferential relative to a continuous metric
If $M$ is a smooth compact orientable manifold without boundary and $g$ a smooth Riemanniann metric on $M$, then one defines the usual $L^2$-inner product on differential forms by
$$\langle \alpha,\...
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Finding the shortest curve that is at distance $\epsilon$ of every point of a surface
Let $M$ be a compact connected (smooth) surface (possibly with boundary) in $\mathbb{R}^3$ and $\epsilon>0$ a constant.
Is there (and if there's not, what conditions on ($M$, $\epsilon$) should ...
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Riemannian foliations and their leaf space
Let be $(M,g)$ a riemannian manifold with a singular riemannian $\mathcal{F}$ in $M$, see [1] the definition of singular riemannian foliation.
The riemannian metric on $M$ induces a distance on $M$, ...
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Parallel transport in Riemannian manifold induces bounded mapping of vector bundles
Let $X$ and $Y$ be closed Riemannian manifolds and $f,g\colon X\to Y$ two $C^1$-mappings.
Assume that for every $x\in X$ the points $f(x)$ and $g(x)$ can be joined by a unique shortest geodesic of $...
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On the Cheeger's estimate of injectivity radius
EDIT: The Cheeger's theorem says that if $M^n$ is a compact smooth Riemannian manifold such that the absolute value of its sectional curvature is less than $\kappa$, diameter at most $D$, and volume ...
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Geodesic symmetry of a locally symmetric space
Let $M = \Gamma \backslash G/K$ be a Riemannian locally symmetric space, where $G$ is a connected semisimple Lie group of rank at least $2$, $K$ its maximal compact subgroup and $\Gamma < G$ an ...
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Integrability of direct sum of some integrable distributions
Let $M$ be a smooth manifold and let $\Delta _i$ for $i=1,...,k$ be distributions of $TM$ which are integrable such that $\Delta_i \cap \Delta _j$ is zero distribution for $i \neq j$. Suppose $\Delta ...
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A question on Cheeger-Gromov compactness theorem
The Cheeger-Gromov compactness theorem says the following. Let us fix $n\in \mathbb{N}$ and positive constants $K,D,v$. Let $\{(M_i^n,g_i)\}$ be a sequence of closed infinitely smooth $n$-dimensional ...
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Regularity of a generalized polar coordinate metric with two angles
Flat space in polar coordinates takes the form
$$ds^2=dr^2+r^2d\phi^2$$
To avoid a conical singularity at the origin, we must impose that $\phi$ is periodic with period $2\pi$.
Now consider the ...
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Manifolds with positive sectional curvature
In the study of manifolds with positive sectional curvature, I guess the following statement is true:
Let $M$ be a manifold with positive (but not necessarily constant) sectional curvature. Then $M\...
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There is no arcwise isometry from a high dimensional manifold into a low dimensional manifold
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Let $X,Y$ be Riemannian manifolds, such that $\dim(X) > \dim(Y)$.
I am trying to prove the following statement (...
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The analyticity of distance function
Given a real analytic compact manifold $M$ with boundary $\partial M$, suppose that $M$ is embedded in an open analytic manifold $N$ which has the same dimension as $M$. Is the distance function $d(x)...
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Does any surface of constant curvature admit a cocompact group action?
Suppose $S$ is a non-compact and complete surface (2 dimensional smooth Riemannian manifold) of constant curvature. I am wondering if there exists a group $G$ which acts by isometries and properly ...
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Conditional distributions of uniformly distributed random orthonormal matrices
Let $U, U'\in R^{d\times k} (d>k)$ be two independent uniformly distributed random orthonormal matrices. In specific, let $S$ be the set of all $d\times k$ orthonormal matrices. Here 'uniform' is ...
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Gromov Hausdorff limit and Ricci flow
Let $M$ be a compact, smooth manifold and $\{g(t)\}$ be a family of Riemannian metrics on $M$ evolving under Ricci flow. Suppose the maximal existence time $T$ is finite. To what extent the following ...
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Property of flat affine connection
Let $M$ be a smooth simply-connected manifold. Let $\nabla$ be a flat, symmetric connection on $M$. Let $p\in M$ and let $v,w\in T_pM$ belong to a normal neighborhood, such that the $\nabla$-geodesic ...
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A careful roadtrip from locally symmetric spaces to algebra
I'm trying to break the classification of locally riemannian symmetric spaces to little steps to make it more comprehensible (and s.t. the technical details can be verified without drowning completely)...
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When is a flow geodesic and how to construct the connection from it
Let $(M,\Gamma)$ be a $C^\infty$ $n$ dimensional real manifold with a linear connection $\Gamma$ on it. I know the following:
If $\gamma:[t_0,t_1]\rightarrow M$ is a smooth curve and is a geodesic, ...
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Entropy degeneration and volume expansion
This question concerns the asymptotic geometry of a sequence of Riemannian metrics on a closed surface whose volume entropies converges to zero.
Let $\Sigma$ be a closed, orientable, connected ...
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Lie bracket of gradients
Related to Why symplectic geometry gives Poisson geometry
Is there any common use for the Lie bracket of the gradients of two functions? That is, $[\nabla f, \nabla g]$?
Here, $\nabla f$ denotes the ...
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Functional inequality under mean curvature flow
Let $\Sigma$ be a hypersurface in $\mathbb R^n$ and $\Sigma_t$ be a variation of $\Sigma$ under the mean curvature flow under an extra condition that ${\rm vol}_{n-1}(\Sigma)={\rm vol}_{n-1}(\Sigma_t)$...
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Riccati equation and principal curvatures
Let $\Omega$ be an open subset of a Riemannian manifold $M$. Assume that $\Sigma:=\partial \Omega$ is $C^2$.
Let $U$ be a neighborhood of $\Omega$ such that $\exp_p(t\nu(p))$ is diffieomorphism, ...
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Geometry of convex sets in Riemannian manifolds
Let $M$ be a smooth Riemannian manifold without boundary. Let $X\subset M$ be a closed subset which is a smooth submanifold with boundary, $\dim X=\dim M$. Assume that $X$ is locally convex, i.e. any ...
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Is there a Bishop-Gromov inequality for manifolds with boundary?
EDIT. Let $M^n$ be a smooth compact Riemannian manifold with smooth boundary.
Assume in addition that near the boundary $M$ is locally geodesically convex.
Assume that the Ricci curvature satisfies $...
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