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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Is this a manifold of bounded geometry?
Let $M$ be a complete non-compact manifold (possibly with boundary). Let $E$ be an open proper connected non-precompact subset of $M$ with smooth topological boundary, so that $\overline{E}$ is a non-...
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Negative sectional curvature and constant curvature
Good morning everyone,
I was wondering about the difference between manifolds carrying a Riemannian metric with negative sectional curvature and hyperbolic manifolds. I was told once "there are ...
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Want a minimal subgroup, whose orbits cover a submanifold, that is contained in a maximal subgroup which leaves the submanifold invariant
Say we have a Lie group $G$ acting transitively on smooth manifold $M$ and take a submanifold $S\subseteq M$. It seems to me that there should be some minimal subgroup $G'\subseteq G$ such that $S\...
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Gap phenomenon vs Rigidity results for surfaces
I am trying to understand the differences between the rigidity results and gap results for a given surface immersed into some manifold. For instance, a Gap theorem proved here (Theorem 2.7) says ...
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Identifying a $4$-form on a $6$-dimensional manifold
Let $M$ be a closed $6$-dimensional Riemannian manifold with a spin$^{\mathbb{C}}$ structure. It is known that real $4$-forms on $M$ act on the positive-spinors as trace-free hermitian endomorphisms ...
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Signs of curvatures of integrals lines of frames with constant principal values
Let $D\subset\mathbb{R}^2$ be a planar domain (maybe simply connected) and consider all the mappings $f:D\to\mathbb{R}^2$ with constant, fixed, positive singular values. Let $E=(E_1,E_2)$ be the ...
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Can you compute one eigenspace without computing them all?
Maybe the simplest non-trivial settings in which the spectrum of the Laplacian be can be computed is on the round sphere $\mathbf{S}^n$, and for products of manifolds. I want to use the two as ...
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Hausdorff dimension of a compact Lie group [closed]
Let $G$ be a compact Lie group (for simplicity assume $G=SO(3)$). Equip $G$ with a left-invariant Riemannian metric and let $m$ be left-invariant Haar measure on $G$.
Now that $G$ is a metric space ...
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Flapping wings: on a question of Kapouleas
The Lawson minimal surfaces $\xi_{1,g} \subset \mathbf{S}^3$ are minimal surfaces with genus $g$. In Lawson's original construction [Law70]
these were constructed from geodesic triangulations. An ...
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Non-compact Dirichlet fundamental domains and free Fuchsian groups
Let $G$ be a finitely generated Fuchsian group, and let $\mathcal{F}$ denote the Dirichlet fundamental domain of $G$ with respect to $0$ in the Poincaré disc model.
Assume throughout that $\mathcal{F}$...
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Superconnected spaces
Question 1. Let $\epsilon > 0$ and $V > 0$. Is there always a complete connected Riemannian manifold $M$ with
$$
\operatorname{diam} M < \epsilon\quad\text{ (small diameter)} \quad \text{and} ...
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Estimating scalar curvature by norm of Riemannian curvature tensor under the Ricci flow
In B. Chow and D. Knopf's book "The Ricci Flow: An Introduction", the authors claim that for any dimension $n$ and any Riemannian manifold $M^n$, there is a constant $C_n$ depending only on $...
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Continuity of the perimeter of level sets w.r.t. level function
Working with the level set method introduced by Osher & Sethian in shape optimization I came across a simple question that I did not succeed to prove. It mainly asserts that the perimeter of the ...
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Hypoellipticity of a heat-like parabolic operator on Riemannian manifolds - reference request
Let $(M,g)$ be a Riemannian manifold and $L$ be a differential operator on $M$, with smooth coefficients, such that its symbol be $g$ (a "generalized Laplacian").
Where can I find proved ...
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Classifying singularities of the Ricci flow
Context:
A solution $(M^n, g(t))$ of the Ricci flow is said to encounter a Type III Singularity if $g(t)$ is defined for all $t \geq 0$ and:
$$
\sup _{\mathcal{M}^{n} \times[0, \infty)} \|\...
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Is there an explanation of analogies between the cross-ratio and the Riemann curvature tensor?
Define the cross-ratio of four real or complex numbers as follows:
$$[a,b,c,d] = \frac{(a-c)(b-d)}{(a-d)(b-c)}.$$
Then its logarithm has the same symmetries as the curvature tensor:
$$\log[a,b,c,d] = -...
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Yau proof of $K_X>0$ using a non-smooth metric which restricts to a metric of negative holomorphic sectional curvature on all curves
In this lecture of Yau's on the Existence of complete Kähler-Einstein metrics with negative scalar curvature he mentions the following, I quote:
Negative holomorphic sectional curvature is a rather ...
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Are there examples of Einstein manifolds with unbounded curvature?
Are there any known examples of Einstein manifolds $(M, g)$ such that $$\sup_{x \in M} \|\text{Rm}(x) \| = \infty$$
I'm looking for these examples because they might provide a counter-example to a ...
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How to compute the Gromov-Hausdorff distance between spheres $S_n$ and $S_m$?
Can we compute the Gromov-Hausdorff distance $d(\mathbb{S}_n,\mathbb{S}_m)$ for two different spheres $\mathbb{S}_n$ and $\mathbb{S}_m$, $m\neq n$? We consider the spheres with the metrics induced by ...
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Continuity of surface integrals on level sets
Let $\phi:\mathbb{R}^2\to\mathbb{R}$ such that $\phi^{-1}(0)\neq\emptyset$ and $\phi\in C^1(W)$ where $W$ is a compact neighborhood of $\phi^{-1}(0)$, with $\nabla\phi\neq 0$ in $W$. So there is some $...
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Levi-Civita connection from idempotents
Let $(M,g)$ be a closed Riemannian manifold. Let $V$ be a smooth complex vector bundle over $M$. We can write $V$ as the range of an idempotent $E$ in a matrix algebra $M_n(C^\infty(M))$ acting on a ...
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Searching for a type of geometric flow in Lorentzian geometry
Let $(N,g)$ be a globally hyperbolic Lorentzian manifold. Given any smooth hypersurface $\Sigma$ in $(N,g)$ we define $\|\Sigma\|= \sup_{p \in N,X \in T_p\Sigma} |h(X,X)|$ where $h$ is the second ...
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Asymptotics of eigenvalues of first-order self-adjoint elliptic operators
Let $D$ be a first-order self-adjoint elliptic operator on a closed Riemannian manifold $M$. Then $D$ has discrete spectrum in $\mathbb{R}$, and there is an orthonormal basis for $L^2(M)$ consisting ...
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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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Does this Banach manifold admit a Riemannian metric?
First, the question; after, the motivation.
Consider 27.6 (pdf pp. 262-263) in The convenient setting of global analysis (AMS, 1997), and, in particular, the example given at the end of it, which ...
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Are there explicit canonical metrics on spheres which are not round?
On a manifold diffeomorphic to $\mathbb{S}^n$, the most canonical metric is the round one. It is also known that there are a large number of Einstein metrics on spheres which are not round. Are there ...
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Explanation of perpendicularity of a Jacobian vector field
Here are some notes on hyperbolic manifolds. The aim is to prove that if $M_1$ and $M_2$ are simply connected, complete Riemannian manifolds having constant sectional curvature of $-1$, then $M_1$ and ...
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Continuity of Hausdorff measure on level sets
Let $\Omega\subset\mathbb{R}^2$ a open and bounded set with smooth boundary and $\phi:\Omega\to\mathbb{R}$ a smooth function such that:
$\bullet$ $\phi^{-1}(0)\neq\emptyset$;
$\bullet$ $\nabla\phi(x)\...
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For globally conformally flat surfaces, does radial symmetry of conformal factor imply the surface is a sphere?
We know that for the unit sphere in $\mathbb{R}^3$, the standard metric on such a sphere can be written as $$g=\frac{4}{(1+x_1^2+x_2^2)^2}(dx_1^2+dx_2^2)$$by a stereographic projection map from the ...
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Birman-Series for variable negative curvature
A famous theorem of Birman and Series says that if $S$ is a compact hyperbolic surface, then the set of points that lie on simple geodesics is nowhere dense and has Hausdorff dimension one; in ...
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Beurling’s extremality criterion for curves: two versions
I see Beurling’s extremality criterion at two places: the proof is almost identical, but the statement is very different. Below,
$$
\ell_\rho (\gamma) = \int_\gamma \rho(z) |dz|.
$$
"Extremal&...
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Why do we always need the Schwarz lemma when bounding the trace of a Kähler metric?
I posted this question on MSE, and while it has received some upvotes, it is not getting much attention. Perhaps it is more relevant here?
My undergraduate thesis topic is Kähler geometry. The general ...
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Ricci curvature of the Weil-Petersson metric?
Let $\omega_{\text{WP}}$ denote the Weil-Petersson metric associated to a family of Calabi-Yau manifolds. That is, let $f : X \to Y$ be a surjective holomorphic map with connected fibres such that, ...
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Critical points of the area functional restricted to CMC embeddings
For fixed closed smooth manifolds $M^n$ and $N^{n+1}$, two $C^{k,\alpha}$ embeddings $f, f' : M \to N$ are said to be equivalent if there exists $\varphi \in \operatorname{Diff}(M)$ such that $f' = f \...
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Alternative to well-known trace estimate in Riemannian geometry?
Let $g,\hat{g}$ be two Riemannian metrics with volume forms $dv_g$, $dv_{\hat{g}}$, respectively. A standard estimate in the subject is the following: $$\text{tr}_g(\hat{g}) \leq \text{tr}_{\hat{g}} (...
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Invertibility of neural network as operator on Wasserstein space
Question statement: Consider the space of probability measures with finite second moments $P_2(\mathbb{R}^d)$, which is equipped with the Wasserstein-2 distance $W_2$, and the square integrable ...
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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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Characterization of planar domains onto which a unit disk can be mapped with constant singular values
It can be shown that there are (smoothly bounded, Jordan) domains $E\subset \mathbb{R}^2$ which are $\textit{not}$ images of mappings $f$ from the unit disk (or any other planar domain), such that $\...
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The isometry group of a product of two Riemannian manifolds
Under what conditions is the isometry group of a product of two Riemannian manifolds the product of the isometry groups of each one of the components?
One counterexample is a product of two isometric ...
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Is every minimal graph smooth?
The following result was taken from the book of Gilbarg-Trudinger:
In particular, if the graph is minimal, then $u$ is smooth.
Now comes my question: does the same conclusion hold for graphs over ...
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Is the Cayley distance on permutation (matrices) equivalent to the Riemannian metric on $O(n)$?
Denote by $d_C(\sigma,\mu)$ the minimal number of transpositions needed to go from a permutation $\sigma$ to a permutation $\mu$. E.g. if $d_C(\sigma,\mu)=0$, then $\sigma=\mu$, if $d_C(\sigma,\mu)=1$,...
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reference to a theorem about a product of harmonic and parallel forms
Let $\alpha$ be an exterior product of a harmonic and a parallel form on a Riemannian manifold. Then $\alpha$ is known to be harmonic. I have heard that this is an old result due to R. Bott, but I ...
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A question on Levi-Civita connection and a fixed hyper surface
Suppose $(M,g)$ is a three dimensional smooth compact simply connected Riemannian manifold with boundary and suppose that $\Sigma$ is a smooth simply connected hypersurface in $M$ with a smooth ...
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Minimal normal graph
Let $(M^3,g)$ be a complete and orientable Riemannian $3$-manifold and let $\Sigma^2 \subset M$ be a compact orientable totally geodesic surface embedded in $M$. For $f \in C^{2,\alpha}(\Sigma)$ with ...
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Immersion of a part of the hyperbolic plane in $\mathbb{R}^3$
I know that the pseudosphere is a regular surface with Gaussian curvature $-1$ that is not complete, also this surface is not complete. Hilbert's theorem ensures that there is no isometric immersion ...
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Isoperimetric inequality for domains in the exterior of a precompact open set in Riemannian manifold
Fix $n\geq 2$ and let $$\mathbb{H}^{n}=\mathbb{R}_{+}\times \mathbb{S}^{n-1}$$ be the hyperbolic space, so that any point $x\in \mathbb{H}^{n}$ can be represented in polar coordinates $x=(r, \theta)$, ...
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A question on null geodesics in Lorentzian geometry
Suppose that $(M,g)$ is a smooth Lorentzian manifold, that $I\subset \mathbb R$ is a closed finite interval and that $\gamma: I \to M$ is a smooth null geodesic on $M$, that is to say, it satisfies
$$ ...
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References Request: Bach tensor
Recently I want to study about Bach tensor in detail. From the fundemental definition and properties to conformal invariant. Is there any references to me about Bach tensor? If there are some origins ...
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Can one extend a Hermitian bundle from a compact manifold with boundary to its Riemannian double?
Let $M$ be a compact Riemannian manifold with boundary, and let $E \to M$ be a Hermitian vector bundle, endowed with a compatible connection. Let $\tilde M$ be a Riemannian double of $M$.
Does $E$ ...
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On Gromov's proof of the systolic inequality $\operatorname{Sys}_1(M)\leq 6\operatorname{FillRad}(M)$
In the page 10 of the paper "Filling Riemannian manifolds" by Gromov (ProjetEuclid link), the author proves the following inequality (1.2) relating the systole and the filling radius of manifolds.
$$\...
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