All Questions
Tagged with dg.differential-geometry riemannian-geometry
1,985 questions
7
votes
2
answers
335
views
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
$$
\operatorname{Sym}^2_0(T^*M) \simeq \Lambda_- \otimes \...
- 108k
2
votes
1
answer
132
views
Existence of Kähler Metric of Bounded Geometry on the Hermitian Vector Bundle on Projective Spaces
A Riemannian manifold $(M,g)$ is said to be of bounded geometry if the Riemannian curvature tensor and its derivatives are bounded, and it has positive injectivity radius.
I am working with the ...
- 129
0
votes
0
answers
32
views
Request for resources on directional derivative of the Riemannian distance function, and Berger's lemma about geodesics realizing the diameter
I've been recently interested in directional derivatives of the Riemannian distance function, and I came across this question, and its answer by Sergei Ivanov, where he stated an important result: (I ...
- 869
1
vote
0
answers
44
views
What do we know about Poisson boundaries of arbitrary Riemannian manifolds?
For closed manifolds, we know that the Poisson boundary is trivial due to compactness and for radially symmetric manifolds for which diffusion is one dimensional, there are A Brief Introduction to ...
- 168
10
votes
1
answer
604
views
Is it possible to average a riemannian metric over an action and preserve curvature bounds?
Let $M$ be a finite dimensional smooth manifold endowed with a riemannian metric $g$ and a smooth action $\mu$ by a compact Lie group $G$. Averaging $g$ over $G$ defines a new metric
$$g'(X,Y)=\int_Gg(...
- 45k
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-...
- 7,817
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 $Sym^m$ ...
- 4,787
3
votes
2
answers
247
views
Morse approximation with bounded number of critical points
Let $(M^3,g)$ be a compact Riemannian 3-manifold and let $f\in C^{\infty}(M)$ be a smooth function. Does there exist a constant $k>0$ (possibly depending on $M$ and $g$) such that $f$ can be $C^2$-...
- 716
4
votes
0
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 $\...
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 ...
- 63.9k
9
votes
1
answer
429
views
Perturbing metrics with nonpositive curvature
Let $M$ be a compact $3$-dimensional manifold diffeomorphic to a ball. Suppose that $M$ has nonpositive (sectional) curvature and its boundary $\partial M$ is convex, or even that $M$ is a Riemannian ...
CommunityBot
- 1
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 ...
- 108k
6
votes
1
answer
604
views
When is the cut locus a finite tree?
Let $\Omega \subset \mathbf{R}^2$ be a bounded, simply connected domain, with a regular boundary, say of class $C^2$ at least. Let the cut locus $C$ of $\Omega$ be the set of points $x \in \Omega$ for ...
CommunityBot
- 1
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 ...
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 ...
0
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.
$$f''(r) + \beta \coth(r) f'(r) = \rho_0 e^{-...
- 272
1
vote
1
answer
144
views
An application of min-max characterization of eigenvalues
Let $(M,g_0)$ be a $n$-dimensional closed Riemannian manifold with a Riemannian covering $(\widetilde{M},\widetilde{g}_0)$. Let
$$
\mathcal{V}_{ab}=\{g\colon a^2 g_0\leq g\leq b^2 g_0\}, \quad \text{...
- 6,001
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
$$\text{Per}_g(\Omega) = \sup\bigg\{\int_{\Omega} \text{div}_g(T) dVol_g, \; : \; T \in C^1(M, T M), \quad \...
CommunityBot
- 1
2
votes
1
answer
172
views
Parallel Jacobi fields in a Hadamard manifold
Let $M$ be a Hadamard manifold and let $c: \mathbb{R}\rightarrow M$ be a geodesic. A Jacobi field $Y$ along $c$ is called parallel if $Y'(t) = 0$ for every $t\in \mathbb{R}$. If we assume that $M$ is ...
CommunityBot
- 1
2
votes
1
answer
326
views
Sectional curvature and injectivity radius of natural metric in cotangent bundles
In the following paper by Cielibak, Ginzburg and Kerman (arXiv link, Comm. Math. Helv. 2004 DOI link) they claim in page $3$ that the natural metric $\tilde g$ on $T^*M$ the sectional curvature is ...
CommunityBot
- 1
1
vote
0
answers
117
views
Question on globally hyperbolic manifolds and coordinates
Consider a globally hyperbolic Lorentzian manifold $(M,g)$. Then, a well-known result of Bernal-Sánchez (see Theorem 1.1 in arXiv:gr-qc/0401112) states that it can globally be written as
$$M=\mathbb{R}...
- 1,171
1
vote
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 ...
- 7,817
20
votes
1
answer
3k
views
The continuity of Injectivity radius
$\DeclareMathOperator{\InjRad}{InjRad}$
Dear all,
when reading a book of M. Berger, I learned that the injectivity radius $\InjRad(x)$ on a compact Riemannian manifold depends continuously on the ...
- 6,367
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
1
vote
1
answer
279
views
On intersection of null geodesics
Let us consider a globally hyperbolic Lorentzian manifold $(M,g)$ with empty cut locus. Suppose that
$p$ is a point in $M$ and consider $C^-(p)$ to be the past null cone in $M$ emanating from the ...
CommunityBot
- 1
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(...
CommunityBot
- 1
4
votes
3
answers
285
views
Bounds on the uniformization map for a metric on the 2-sphere
Suppose $m$ is a smooth Riemannian metric on $\mathbb{S}^2$, the uniformization theorem of surfaces tell us that $m$ is conformally equivalent to the standard round metric. Formally this says that ...
- 68.9k
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 $...
- 63.9k
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
4
votes
3
answers
927
views
Lower bound for the normal injectivity radius
Let $(M,g)$ be a closed Riemannian manifold and let $N$ be a closed embedded submanifold. A tube $T(N,r)$ of radius $r$ of $N$ is defined as the set of points of $M$ which can be reached by a ...
- 1
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
10
votes
0
answers
160
views
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
0
votes
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 ...
- 6,367
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
0
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
3
votes
1
answer
135
views
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 ...
CommunityBot
- 1
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 ...
- 108k
3
votes
1
answer
874
views
Motivation and examples of parabolic manifolds
Let $(M^{n},g)$ be a Riemannian manifold, we say that $M$ is parabolic if the constant functions over $M$ are the only subharmonic functions which are bounded above, i.e, for a function $u \in C^{2}(M)...
CommunityBot
- 1
6
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
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,...
- 7,817
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 ...
- 7,817
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 ...
- 35.5k
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 ...
- 63.9k
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
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
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{...
- 63.9k
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 ...
- 28.2k
1
vote
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 $u,v\in ...
- 111
0
votes
0
answers
134
views
Positive mass theorem and Seiberg-Witten equations
Apologies for not a very rigorous question. I came across this PhD thesis by XIAO ZHANG, a student of Yau. From the thesis:
"We also investigate some
basic facts on Spin$^c$ structure on $4$-...
- 954