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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Regularity of the distance from the boundary in singular riemannian manifolds
I am looking for references related with the regularity of the distance from the boundary in singular Riemannian manifolds.
I assume the following setting. $(M,g)$ is a Riemannian manifold, with ...
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Equivariant isometric embedding of manifolds in a Hilbert space under a noncompact group action
Given a Riemannian manifold $M$ and a group of isometries $G$ of $M$, I am interested in when there exists a isometric embedding $\iota : M \to H$, where $H$ is a Hilbert space and a representation $\...
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configuration space of Riemannian manifolds with a parameter on the distance of distinct points
Let $M$ be a Riemannian manifold. For any $\epsilon\geq 0$, we define the $k$-th ($k=1,2,\cdots$) "$\epsilon$-configuration space" as
$$
F(M,k,\epsilon)=\{(x_1,\cdots,x_k)\in M^k\mid d(x_i, x_j)>\...
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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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Toponogov comparison theorem for complex manifold
I would like to know some reference for the Toponogov comparison theorem for complex manifolds, in particular for complex manifolds with bounded holomorphic sectional curvature. As far as I know, the ...
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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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stochastic control / geometric mean
Consider the following problem:
Given $\Omega$ and $U$ two symmetric definite positive matrices, choose a matrix $K$ to minimize the expectation $x' \Omega x + x'K'UKx$ when $x$ follows the invariant ...
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Relationship between geodesics and curvature lines on surfaces?
I'm trying to understand the relationship between geodesics and lines of principal curvature (to keep things simple, let's say Riemannian 2-manifolds embedded in $\mathbb{R}^3$). In my reading, I ...
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A question in spin geometry in dimension 8
$\DeclareMathOperator\trace{trace}\DeclareMathOperator\End{End}\DeclareMathOperator\Trace{Trace}$This is to understand a very specific isomorphism in dimension $8$. In dimension $4$ for a spin$^c$ ...
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Jacobian of exponential map
I am playing around with the coarea formula and came across the problem of finding the Jacobian of the exponential map.
Let $G$ be a compact, semisimple Lie group with associated Lie algebra $\...
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Sequence of 2-cylinders converging to a segment in the Gromov-Hausdorff metric
Let $\{C_i\}_{i=1}^\infty$ be a sequence of (compact) 2-dimensional cylinders with smooth Riemannian metrics with Gauss curvature at least $-1$ and geodesically convex boundary (equivalently, the ...
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Curvature of geodesic spheres
Suppose $\mathbb{R}^d$ is given a Riemannian metric $g$. Fix a point $p \in \mathbb{R}^d$, fix two tangent vectors $v,w \in T_p \mathbb{R}^d$ with $\langle v, w \rangle_g = 0$, and consider the ...
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Mean-value type property for eigenfunctions of Casimir operator on $\operatorname{SL}(2,\mathbb{R})$
The purpose of this post is to ask for the community's opinion on a conjecture about a certain mean-value property for functions on $\operatorname{SL}(2,\mathbb{R})$. This conjecture appears at the ...
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Isometry group of the Fubini-Study metric on complex projective spaces
Let $(\mathbb CP^n,g_{FS})$ be the complex projective space equipped with the standard Fubini-Study metric.
What is the Riemannian isometry group of $(\mathbb CP^n,g_{FS})$? It seems to me that its ...
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Regularity of exponential map for $C^{2,\alpha}$ Riemannian metrics
Let $g$ be a $C^{2,\alpha}$ Riemannian metric and $0<\alpha<1$. Would the exponential map $\mathrm{exp}_p$ be $C^{1,\alpha}$ as the point $p$ varies?
Since $\mathrm{exp}_p$ is defined by the ...
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Parallel transport of global sections and Riemannian curvature
A, perhaps, naive question from an algebraist/combinatorialist teaching differential geometry. Originally asked on math.SE but didn't receive a single comment in 3 days.
Consider a (real) smooth ...
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Trace-class heat semigroups
Let $(M,g)$ be a compact Riemannian manifold and $\Delta_g$ its Laplace operator.
Let $\varphi$ be a test function on $\mathbf{R}_{>0}$. We define the operator on, say, $L^2(M)$
$$T_{\varphi}(u) :=...
user516086
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Dynamical obstruction for a vector field to have a Harmonic divergence
Let $(M,g)$ be an analytic Riemannian manifold and $X$ be an analytic vector field on $M$. Can we always have a volume form $\Omega$ such that $\operatorname{Div}_{\Omega} X$ is a harmonic ...
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Homotopy type / Homology of the free loop space of aspherical manifolds
Let $X$ be a (connected, smooth) closed aspherical manifold. Let $LX:=Map(S^1,X)$ be the free loop space of $X$. Pick $x_0\in X$ and let $\Omega_{x_0}(X)$ be the based loop space of $X$ (based at $x_0$...
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Riemannian manifolds with a unique distance property
Let $M$ be a compact Riemannian manifold with geodesic distance function $d$, of (normalised) diameter $1$.
Some of my favourite manifolds $M$ have the property that there exists an integer $k$ such ...
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Is there a good description of harmonic maps from $\mathbf{C}$ to $\mathbf{H}$?
Given a non-constant holomorphic quadratic differential $\phi \, \mathrm{d} z^2$ on the complex plane $\mathbf{C}$, there is a harmonic diffeomorphism $u: \mathbf{C} \to \mathbf{H}$ into the ...
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Eigenvalues of Schrödinger operator with Robin condition on the boundary
Let $(M^2,g)$ be a compact Riemannian surface with boundary and let $L = \Delta_g + q$ be a Schrödinger operator, where $\Delta_g = -\operatorname{div} \nabla$ is the Laplacian with respect to the ...
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is there a co-Stokes theorem? (with the codifferential)
I am trying to read through J. Simmons, Minimal Varieties in Riemannian Manifolds, and in the proof of Proposition 1.2.2 he calls "Stokes theorem" to the following result:
$$
\int_M\delta\...
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Symmetric line spaces are homeomorphic to Euclidean spaces
For points $x,y,z$ of a metric space $(X,d)$ we write $\mathbf Mxyz$ and say that $y$ is a midpoint between $x$ and $z$ if $d(x,z)=d(x,y)+d(y,z)$ and $d(x,y)=d(y,z)$.
Definition: A metric space $(X,d)$...
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Hodge decomposition on non-compact manifolds
Let $(\mathcal{M},g)$ be a compact Riemannian manifold without boundary. Then we have the well-known Hodge decomposition
$$\Omega^{k}(\mathcal{M})\cong\mathcal{H}^{k}(\mathcal{M})\oplus\mathrm{ran}(\...
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Upper bound for the first eigenvalue of the Laplacian on surfaces with boundary
Let $\Sigma$ be a compact smooth surface with boundary. Define
$$\Lambda(\Sigma) := \sup \{ \lambda_1(\Sigma,g) \operatorname{Area}(\Sigma,g) : g \text{ is a smooth Riemannian metric on $\Sigma$} \}$$
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Representing homotopy classes of Kähler manifolds by harmonic maps
Let $(M,g_M)$ be a compact Kähler manifold with negative bisectional curvature. Let $\alpha : (S,g_S) \to M$ be a continuous map from a compact Riemannian manifold $(S,g_S)$.
Is $\alpha$ homotopic to ...
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Let $p : \tilde{M} \to M$ be the universal cover. Can we ever deduce curvature properties of $M$ from the curvature of $\tilde{M}$?
Let $M$ be a $C^{\infty}$-smooth, connected, paracompact manifold with universal cover $\tilde{M}$. Assume $M$ is not simply connected, so that the covering map $p : \tilde{M} \to M$ is not the ...
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Does the conformal factor satisfy some equation independent of the vector field?
Let $(M, g)$ be an $(n+1)$-dimensional (pseudo-)Riemannian manifold and for any
$X \in \mathcal{X}(M)$ define its deformation tensor by
\begin{align*}
{} ^{(X)}\pi_{ab} := (\mathcal{L}_X g)_{ab} = \...
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Request for two unpublished articles of Detlef Gromoll
I am looking for two articles for my research purpose, which are entitled with "Convex riemannian manifolds" and "Convex sets in riemannian manifolds" by Detlef Gromoll.
I would ...
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To what extent is the Nash embedding not unique?
Consider a smooth Nash embedding, $f$, of a Riemannian manifold $Σ$ into Euclidean space $\mathbb R^n$. To what extent is this embedding not unique?
It is clear that the set of all such embeddings ...
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Good resources that talk about geodesically convex sets for riemannian manifolds?
Are there any good resources that talk about geodesically convex sets, and in particular convex hulls, for riemannian manifolds? I’ve been wanting to learn more about them, their geometry and ...
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Hodge theory in higher eigen-spaces?
Hodge theory for elliptic complexes $E$ identifies the space of harmonic sections with cohomology
$$\mathcal{H}(E) \simeq H(E).$$
A classical example with differential forms ($E = (\Omega,d)$) ...
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On the convergence of the spectral decomposition of a harmonic function
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 2$ with a smooth boundary. Denote by $0<\lambda_1\leq \lambda_2\leq\ldots$ the Dirichlet eigenvalues of $-\Delta_g$ on $(M,g)$...
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Low boundary of $\mathcal W$ function
Picture below is from Topping's Lectures on Ricci flow.
I don't understand the red line. From Lemma 8.1.8, I can get that $\mathcal W (g,f,\tau)$ has low boundary for any compatible $f,g,\tau$. But ...
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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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How many ways are there to characterise $\mathbb{P}^n$?
Let $\mathbb{P}^n$ denote the complex projective space of dimension $n$. In many respects, this is the model of (positivity in) complex geometry. There are some well-known characterisations of $\...
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History of Laplacian comparison theorem
The Laplacian comparison theorem says that if a $n$-dimensional Riemannian manifold has nonnegative Ricci curvature, then the distance function to any point satisfies $\Delta d\leq\frac{n-1}{d}$. ...
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$1$-parameter family of minimal embeddings and the maximum principle
Let $M^3$ be a closed orientable smooth manifold and let $g_t$ be a (smooth) $1$-parameter family of Riemannian metrics on $M$, $t \in \mathbb{R}$. Let $P \subset M$ a fixed closed orientable embedded ...
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The range of $\int_M \kappa_g ds$ where $g$ varies in all possible real analytic metrics on $M$
Let $M$ be a real analytic open surface(A non compact 2 dimensional manifold without boundary).
For every number $\lambda\in \mathbb{R}$, is there a real analytic Riemannian metric on $M$ with
$$\...
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Classifying singularities of the Ricci flow
Context:
A solution $(M^n, g(t))$ of the Ricci flow is said to encounter a Type III Singularity if $g(t)$ is defined for all $t \geq 0$ and:
$$
\sup _{\mathcal{M}^{n} \times[0, \infty)} \|\...
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Products of eigenfunctions on compact Riemann surfaces
Let $M$ be a compact Riemann surface with genus $g\geq 2$, endowed with the Riemannian metric with constant sectional curvature $-1$. Let $f_1, f_2$ be two (global) eigenfunctions for the Laplace-...
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Is the heat semigroup on a manifold the limit of the heat semigroups associated to a compact exhaustion?
Let $M$ be a paracompact Riemannian manifold, and $E \to M$ a Hermitian vector bundle endowed with a Hermitian connection $\nabla$. Write $M$ as an exhaustion $\bigcup _{j \ge 0} U_j$ with relatively ...
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Sequence of minimal surfaces with bounded second fundamental form and area
Let $M^3$ be a closed orientable smooth manifold, let $g_n$ be a sequence of Riemannian metrics on $M$ converging to $g$ and let $\Sigma_n$ be a sequence of closed orientable $g_n$-minimal surfaces ...
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Eigenvalues of Laplacian and eigenvalues of curvature operator
Let $(M^n,g)$ be a compact Riemannian manifold (without boundary). The symmetries of the curvature $R$ of (the Levi-Civita connection associated to) $g$ allow one to realise $R$ as a self-adjoint (...
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Range of divergence operator on the space of traceless symmetric $(0,2)$ tensors; conformal vector fields on an arbitrary metric on $S^2$
Let $\gamma$ be a metric on $S^2$.
I am trying to solve the following PDE on a $(0,2)$ symmetric traceless tensor $A$:
$$div_{\gamma} A = \omega$$
where $\omega$ is a 1-form.
It is known that there ...
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Sobolev space of maps between manifolds with boundary
Let $(M,g)$ and $(N,h)$ be compact Riemannian manifolds with non-empty smooth boundary.
If we consider the Sobolev space $W^{1,p}(M,N)$, is there a reference
on how to model this as a manifold?
If ...
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Isometries of fiber bundles
Let $F\to S\overset{\pi}{\to} B$ a Riemannian submersion with totally geodesic fibers.
Question: How much information about the isometries of $S$ we have if we know the isometries of $F$ and $B$? For ...
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Umbilic points of minimal hypersurfaces and distributional Simons inequality
Let $\Sigma$ be a minimal hypersurface of a smooth Riemannian manifold $(M,g)$ with second fundamental form $h$. What can one say about the set $\{p\in\Sigma:h(p)=0\}$? Is each point isolated? (I feel ...
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An inequality about metric spaces
I started studying this article(《$L^2$ CURVATURE BOUNDS ON MANIFOLDS WITH BOUNDED RICCI CURVATURE》) about 3 months ago: arxiv.org/abs/1605.05583
In this article, there is a seemingly simple assertion ...
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