All Questions
Tagged with dg.differential-geometry riemannian-geometry
1,986 questions
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Request for resources on directional derivative of the Riemannian distance function, and Berger's lemma about geodesics realizing the diameter
I've been recently interested in directional derivatives of the Riemannian distance function, and I came across this question, and its answer by Sergei Ivanov, where he stated an important result: (I ...
- 1,467
1
vote
0
answers
44
views
What do we know about Poisson boundaries of arbitrary Riemannian manifolds?
For closed manifolds, we know that the Poisson boundary is trivial due to compactness and for radially symmetric manifolds for which diffusion is one dimensional, there are A Brief Introduction to ...
- 168
-2
votes
0
answers
66
views
Interplay Between Curvature, Volume, and Optimization: Extending Results Beyond Surfaces of Revolution in $\mathbf R^3$ [closed]
Recently, I discovered a precise formulation directly relevant to my research:
Let $X=[0,1]^n$. For all $n>2$ does $X$ admit a unique codimension one surface of revolution, $L$, with a complete ...
- 105
7
votes
2
answers
334
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Reference request: $\operatorname{Sym}^2_0(T^*M) \simeq \Lambda_- \otimes \Lambda_+$
I am looking for proof of the "well-known" result that for a $4$-dimensional Riemannian manifold $(M, g)$, we have an isomorphism
$$
\operatorname{Sym}^2_0(T^*M) \simeq \Lambda_- \otimes \...
- 113
4
votes
1
answer
547
views
Question on Lorentzian geometry
I apologize in advance if this is a too basic question.
Let $(M,g)$ be a Lorentzian manifold with signature convention $(-,+,\dots,+)$. Now, lets suppose $X\in\Gamma(TM)$ defines a global time-...
- 1,171
2
votes
1
answer
131
views
Existence of Kähler Metric of Bounded Geometry on the Hermitian Vector Bundle on Projective Spaces
A Riemannian manifold $(M,g)$ is said to be of bounded geometry if the Riemannian curvature tensor and its derivatives are bounded, and it has positive injectivity radius.
I am working with the ...
- 21
3
votes
1
answer
213
views
Geodesic flows and Killing fields
How well-known is the following result:
Let $(M,g)$ be a closed Riemannian manifold and define $D:C^\infty(M,Sym^mT^*M)\to C^\infty(M,Sym^{m+1}T^*M)$, $D\alpha:=\mathbb S(\nabla\alpha)$ where $Sym^m$ ...
1
vote
1
answer
158
views
Comparison of special metrics on Riemann surfaces with the hyperbolic one
Let $X$ be a complex algebraic curve, or equivalently a compact Riemann surface, of genus $a\ge2$. Denote $\omega_1, \dots , \omega_a$ a basis of holomorphic differentials, and consider the Riemaniann ...
- 31
3
votes
1
answer
257
views
Asymptotic parametrization for negatively curved surfaces
Let $S$ be a complete simply connected negatively curved surface immersed in Euclidean space $\textbf{R}^3$. Does there exist a parametrization $f\colon\textbf{R}^2\to\textbf{R}^3$ for $S$ such that ...
- 7,149
2
votes
0
answers
70
views
Laplace spectrum on $U(n)$
Consider $\psi:SU(n)\times U(1)\to U(n)$, $(w,z)\mapsto \bar{z}\cdot w$. One can show that $\psi$ serves as a projection and $SU(n)\times U(1)$ is a principal $\mathbb Z_n$-bundle over $U(n)$.
Suppose ...
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votes
0
answers
76
views
Existence solutions of the system of equations on Riemannian manifold
Is there a way to show that the following system of two equations has a solution? I don't want to find an explicit solution, but just verify its existence.
$$f''(r) + \beta \coth(r) f'(r) = \rho_0 e^{-...
- 272
3
votes
1
answer
136
views
$L^\infty$-bound on Laplace-eigenfunctions
Suppose we are on a closed Riemannian manifold $M$. Any function $f\in C^\infty(M)$ may be decomposed as $$f = \sum_{j = 0}^\infty f_j\phi_j,$$ where $\phi_j\in C^\infty(M)$ are the Laplace ...
4
votes
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answers
236
views
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 $\...
- 133
1
vote
1
answer
144
views
An application of min-max characterization of eigenvalues
Let $(M,g_0)$ be a $n$-dimensional closed Riemannian manifold with a Riemannian covering $(\widetilde{M},\widetilde{g}_0)$. Let
$$
\mathcal{V}_{ab}=\{g\colon a^2 g_0\leq g\leq b^2 g_0\}, \quad \text{...
- 563
1
vote
0
answers
117
views
Question on globally hyperbolic manifolds and coordinates
Consider a globally hyperbolic Lorentzian manifold $(M,g)$. Then, a well-known result of Bernal-Sánchez (see Theorem 1.1 in arXiv:gr-qc/0401112) states that it can globally be written as
$$M=\mathbb{R}...
- 1,171
1
vote
0
answers
30
views
Critical point of perturbed stratifiable function has no cluster point
Given a smooth function $f:\mathbb{R}^d\rightarrow\mathbb{R}$ and a smooth manifold $\mathcal{M}$. Now, consider the set
$$
S(v)=\{x:0\in x\circ(\nabla_{\mathcal{M}} f(x)+N_{\mathcal{M}}(x)+v)\}.
$$
...
- 43
6
votes
2
answers
390
views
Continuity of perimeter with respect to metric
Let $\Omega$ be an open set in a closed manifold, $(M^n, g)$. We can define the perimeter as
$$\text{Per}_g(\Omega) = \sup\bigg\{\int_{\Omega} \text{div}_g(T) dVol_g, \; : \; T \in C^1(M, T M), \quad \...
- 337
1
vote
1
answer
195
views
The length is bounded
Let $\Sigma$ be a surface of finite type. Let $\mathcal{S}$ be the set of non-trivial isotopy classes of simple closed curves on $\Sigma$. One denotes by $l_x(\alpha)$ the infimal length of curves in ...
- 1,043
3
votes
1
answer
189
views
Randomly perturbed function has no accumulated critical point almost surely?
Given a smooth function $f$ and a smooth manifold $\mathcal{M}$ in $\mathbb{R}^d$, define the set
$$
S(v):=\{x:{\rm Proj}_{T_x{\mathcal{M}}}(v)=\nabla_{\mathcal{M}}f(x)\}.
$$
Is correct to say that $S(...
- 43
1
vote
0
answers
81
views
Quotients of the Hilbert space
Let $G$ be a compact Lie group with a biinvariant metric.
Note that $G\times G$ acts isometrically on $G$ from left and right.
Consider the quotient $D=G/H$ by a closed subgroup $H\le G\times G$;
if $...
- 45k
3
votes
0
answers
72
views
Compactness of bounded index solutions of the Yamabe problem
Consider, a closed Riemannian manifold $ (M^n,g) $ , $ n \geq 3 $, with positive Yamabe invariant: $$ 0< Y(M, [g]):= \inf_{0<v \in H^1} Q_g(v), $$ where $$ Q_g(v) = \inf_{0 <v \in H^1} \...
- 457
5
votes
0
answers
78
views
Is there a generalization of the Diameter Sphere Theorem to orbifolds?
The Diameter Sphere Theorem of Grove and Shiohama asserts that if $M$ is a compact Riemannian manifold with sectional curvature bounded from bellow by 1 and diameter greater than $\pi/2$, then $M$ is ...
- 51
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0
answers
160
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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 $\partial S=\gamma$?
- 2,934
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0
answers
141
views
The tensor product of two Fredholm operators
What can be said about the tensor product $T\otimes S$ of two Fredholm operators $T:X_1\to Y_1$ and $S:X_2 \to Y_2$ where $X_1,X_2,Y_1, Y_2$ are Banach spaces and tensor product of ...
- 356
0
votes
0
answers
65
views
Regularity of Metric when defining C^k norms
Given $(M^n, g)$ closed riemannian manifold, I am wondering about the definition of the $C^k$ norms with respect to the metric, and how these norms depend on $g$. For example, I would assume that if $...
- 337
2
votes
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answers
172
views
Second variation formula in Spivak, Volume 4
Let $\alpha: (-\varepsilon,\varepsilon) \times [0,1] \to M$ be a smooth variation of geodesics on a Riemannian manifold $M$, not necessarily fixed at endpoints. Then in Spivak, Volume 4, Chapter 8, ...
- 455
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votes
1
answer
135
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Geodesic convexity of Dirichlet Fundamental Domains
My question is motivated by this question, and this answer to it. Below, let's consider the setup in that answer:
Let $M$ be a Riemannian manifold. Let $G\times M\to M$ be a proper action of a ...
- 1,467
3
votes
1
answer
264
views
How badly does the geodesic exponential map fail to be $C^2$ on Finsler manifolds
Consider a Finsler manifold $M$. Then for each $x \in M$, we can consider the partial map $\exp_x: T_x M \to M$, which is $C^\infty$ away from the origin, $C^1$ at the origin, but never $C^2$ at the ...
- 455
8
votes
0
answers
103
views
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 ...
- 1,429
2
votes
0
answers
72
views
Diameter bounds by mean curvature and area
I'm wondering about a generalization of Simon/Topping/Wu-Zheng's results on bounding diameter by the mean curvature, which roughly says: given a closed $\Sigma^{n-1} \subseteq M^n$,
$$\text{diam}(\...
- 337
-1
votes
1
answer
145
views
Is a Riemannian submersion a harmonic map?
Is every Riemannian submersion necessarily a Harmonic map? If not under what condition that is true?
The motivation: the linear part of a Riemannian submersion is the direct sum og an isometry ...
- 356
2
votes
0
answers
97
views
Property of parallel translation in Green and Wu, "On the subharmonicity and plurisubharmonicity of geodesically convex functions"
In the mentionned paper, I am having difficulties in understanding the proof of lemma 2.
Roughly, this lemma says that given any separation $\eta$ for the $C^\infty$ topology of smooth paths from $[-1,...
- 21
4
votes
0
answers
96
views
Curvature of geodesic spheres
Suppose $\mathbb{R}^d$ is given a Riemannian metric $g$. Fix a point $p \in \mathbb{R}^d$, fix two tangent vectors $v,w \in T_p \mathbb{R}^d$ with $\langle v, w \rangle_g = 0$, and consider the ...
- 455
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vote
0
answers
140
views
Question on harmonic map between Riemannian manifolds
Let $S=[0,1]^2$. Ignoring issues having to do with boundaries and corners, a chart is a diffeomorphism $\varphi \colon S \to S$. Let $g_0$ denote the flat (euclidean) metric. Given a chart $\varphi \...
- 383
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votes
1
answer
334
views
Tensor component calculation
First of all, this question may be more suited for the Math stack exchange site. If anyone finds this question irrelevant here, please transfer to the relevant site.
Recall that in terms of Weyl and ...
- 319
2
votes
0
answers
82
views
Is isoperimetric hypersurface unique up to homeomorphism?
Is there a Riemannian structure on $\mathbb{R}^n $with two non homeomorphic compact hypersurfaces $M,N$ such that both satisfy the isoperimetric inequality. I precisely meanthe following:
$$\...
- 356
1
vote
1
answer
109
views
Horizontal knots on 3 sphere
Motivation: First I present my motivation for this question but this motivational part is not my main question.
I participated in a talk on knot theory. Then I presented the following ...
- 356
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votes
0
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134
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Positive mass theorem and Seiberg-Witten equations
Apologies for not a very rigorous question. I came across this PhD thesis by XIAO ZHANG, a student of Yau. From the thesis:
"We also investigate some
basic facts on Spin$^c$ structure on $4$-...
- 954
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0
answers
122
views
Bilipschitz constants of exponential map on small ball for Riemannian manifold with curvature bounds
Let $(M,g)$ be a Riemannian manifold with sectional curvature $\mathrm{sect}$ between $-K\le \mathrm{sect} \le K$ for some $K>0$. In [1] it is stated at the beginning of section 4, that if $u,v\in ...
- 111
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vote
0
answers
53
views
The Frobenius integrability of distrbution and Hyers–Ulam–Rassias stability
Let $M$ be a compact Riemannian manifold. The norm of vector fields are computed with respect to the metric. Moreover for every distribution $D$, the orthogonal projection on $D$ is ...
- 356
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votes
0
answers
37
views
Does smallness of Gromov-Hausdorff distance on scale 2 imply smallness on GH distance on scale 1?
Let $(M,g)$ be a Riemannian manifold and $C(Y)$ be a metric cone over $Y$. Let $B_r$ denote the geodesic ball of radius $r$ centered at a fixed point $x$ in $M$ and $C_r$ denote the metric ball of ...
- 151
2
votes
0
answers
92
views
Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries
For $n \geq3$, let $(M,g)$ be smooth $n$-dimensional, compact, Riemannian manifold with a smooth boundary. Then there exists some constant $A=A(M,g)>0$ such that, for all $u \in H^1(M)$
\begin{...
- 101
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votes
3
answers
893
views
When are these base spaces isomorphic?
Given a smooth manifold $\mathcal{M}$ that is a fibre bundle over two different base spaces, i.e., there are $\Pi_1:\mathcal{M}\rightarrow B_1$ and $\Pi_2:\mathcal{M}\rightarrow B_2$, if I can prove ...
3
votes
2
answers
236
views
Lengths of closed geodesics and geodesic segments
Let $M$ be a closed Riemannian manifold of dimension $n \geq 2$. I am looking at sufficient conditions for the following two properties:
existence of closed geodesics of arbitrarily long length on $M$...
- 31
6
votes
1
answer
196
views
On elliptic operators on non-compact manifolds
Let $(M,g)$ be a (connected, oriented) Riemannian manifold and $E$ some finite-rank $\mathbb{R}$- or $\mathbb{C}$-vector bundle equipped with some (positive-definite) inner product on the level of (...
- 1,429
4
votes
1
answer
254
views
Isometry group of a left-invariant Riemannian metric on $\mathrm{SU}(2)$
Recall that
\begin{equation}
\mathbb{S}^3=\operatorname{SU}(2)=\left\{
\begin{pmatrix}
z&w\\
-\bar{w}&\bar{z}
\end{pmatrix}
,|z|^2+|w|^2=1
\right\}
\end{...
- 1,441
4
votes
0
answers
114
views
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 ...
- 1,769
3
votes
0
answers
118
views
Decomposition of forms in $\operatorname{SU}(4)$-manifold
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}$Let $(X,\Omega,\omega,J)$ be a manifold with an $\SU(4)$ structure. Since $\SU(4)\subset\Spin(7)$, $X$ also has a $\Spin(7)$-structure. I ...
- 954
6
votes
1
answer
297
views
Understanding exterior differential systems
Let $M$ be an $n$-dimensional smooth manifold. An exterior differential system on $M$ is by definition a graded ideal $\mathcal{I}\subset \Omega^{\bullet}(M)$ in the ring $\Omega^{\bullet}(M)$ of ...
- 2,816
1
vote
0
answers
100
views
Dirac operator on $\operatorname{Spin}(7)$, $G_2$ and $\operatorname{SU}(3)$ manifolds
$\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\SU{SU}$Let's take a $\Spin(7)$ manifold $M$ (the $\Spin(7)$ structure can have torsion), then the standard Dirac operator from negavtive spinors to ...
- 954