All Questions
Tagged with dg.differential-geometry riemannian-geometry
237 questions
11
votes
4
answers
1k
views
Surfaces with non-constant negative curvature
Are there any nice models of surfaces with non-constant negative curvature, analogous to the Poincare disk for constant negative curvature. I have found lots of general results and theory but no nice ...
- 680
11
votes
2
answers
826
views
Nash isometric embedding for noncompact manifolds
It seems that the smooth isometric embedding theorem by Nash is true also for noncompact manifolds.
Is it true that any (complete, connected) Riemannian manifold $(M^n,g)$ admits a proper smooth ...
- 3,146
11
votes
4
answers
3k
views
Laplace-Beltrami Operator on Surfaces
I would like to know what is known about the spectrum of the Laplace-Beltrami operator on 2-dimensional negatively curved surfaces of constant curvature.
For instance,
What is the spectrum of the ...
- 3,626
11
votes
1
answer
1k
views
Does the gluing procedure in Robert Wald’s book *General Relativity* yield a Hausdorff spacetime?
Before I state my problem, let me provide some definitions pertaining to the Cauchy Problem in General Relativity.
Definition 1: A triplet $ (\Sigma,h,k) $ is called an initial data set if $ (\Sigma,...
- 307
10
votes
1
answer
848
views
Boundary terms of formal adjoints of differential operators
Let $M$ be a compact manifold with boundary. If we have two vector bundles $E, F \to M$ with inner products and a differential operator $D: C^{\infty}(E) \to C^{\infty}(F)$ then $D$ admits a formal ...
- 1,601
10
votes
3
answers
721
views
Number of disjoint simple closed geodesics
According to Jairo comment on the first version of this question I revise the question as follows;
Let $g$ be a real analytic Riemannan metric on $S^{2}$. Is it true to say that:
There are at most a ...
- 356
10
votes
4
answers
710
views
Palais's and Kobayashi's theorems on automorphism groups of geometric structures
My question concerns two results in the neighborhood of the standard theorem of Myers-Steenrod that isometry groups of Riemannian manifolds are Lie groups. Both appear in the first chapter of ...
- 486
10
votes
1
answer
2k
views
Global description of the Levi-Civita connection
I'm interested in finding a global (coordinate-free) description of the Levi-Civita connection on a (possibly infinite-dimensional) Riemannian manifold X.
I'm not looking for a description of this ...
- 8,803
10
votes
2
answers
862
views
Deforming metrics from non-negative to positive Ricci curvature
Given a closed Riemannian manifold $(M,g)$ with non-negative Ricci curvature and $dim\geq 3$, when can we deform the metric to a positive Ricci curved one?
I know it's impossible in general due to ...
- 123
10
votes
1
answer
265
views
Unique factorisation of prime geodesics?
In T. Sunada's 1985 paper ``Riemannian coverings and isospectral manifolds'', it is noted that for a compact Riemannian manifold $M$, prime (closed and non self-intersecting) geodesics behave like ...
- 5,701
9
votes
2
answers
2k
views
Distance function to a submanifold
Let $M$ be a compact Riemannian manifold and $\Sigma\subset M$ a closed submanifold. Given $x\in M$ we define the distance function to $\Sigma$ by $$d_\Sigma(x):=\inf\{d(x,y):y\in \Sigma\},$$ where $d$...
- 443
9
votes
1
answer
1k
views
How submanifolds evolve under Ricci flow?
This may be very naive, since I just started trying to learn Ricci flow; but I couldn't really find any answer after looking for a while in all the textbooks and lecture notes I found online...
...
- 5,891
9
votes
2
answers
2k
views
$J$-holomorphic curve as a minimal surface
The following is a part of the proof of Gromov nonsqueezing theorem.
The existence of a $J$-holomorphic curve gives an upper bound for the radius of a symplectically embedded ball.
Let $\psi: B(r) \...
- 1,398
9
votes
1
answer
366
views
Smooth isometric embedding of euclidean n-space into an arbitrarily small neighborhood of another euclidean space
It is not hard to isometrically embed the euclidean plane smoothly into an arbitrarily small neighborhood of euclidean 4-space: First embed a real line isometrically in an arbitrarily small ...
- 2,858
9
votes
1
answer
838
views
Conformal changes of metric and geodesics
Suppose $(M,g)$ is a Riemannian manifold. Let us assume that $X$ denotes a vector field in this manifold and consider the integral curves of this vector field.
Does there exist a conformal factor $c$ ...
- 4,135
8
votes
1
answer
357
views
Estimates of $\Delta|\nabla u|$ for harmonic function $u$
The well-known Bochner formula and refined Kato inequality (for harmonic function) tells us that for a harmonic function $u$ in $B_{2}\subset\mathbb{R}^{n}$,
$$
\frac{1}{n}|\nabla^{2}u|^{2}\leqslant |\...
- 155
8
votes
2
answers
471
views
Obstructions for the wedge of coordinate differentials to be harmonic
Let $(M,g)$ be a smooth $d$-dimensional Riemannian manifold, $d$ even. Are there obstructions (I guess in terms of curvature) for $g$ to have the following property:
For every $p \in M$ there exist a ...
- 6,741
8
votes
0
answers
191
views
Lifting a determinant map
This is a kind of a follow-up to Question on Hessian of a function (probability question). Suppose I give you a continuous function $f:\mathbb{R}^n \to \mathbb{R}.$ Is it true that there exists a ($C^...
- 96.4k
8
votes
3
answers
1k
views
Examples of manifolds that do not admit scalar flat metrics
The Kazdan-Warner trichotomy states that for $n\ge 3$, a compact $n$-manifold falls into one of three categories:
(A) Every (smooth) function is a scalar curvature.
(B) The manifold is strongly ...
- 738
8
votes
1
answer
673
views
Classification of compact globally symmetric spaces
It is known that any connected compact Lie group $G$ is a finite quotient of the product of a compact simply connected semisimple Lie group $\tilde{G}$ and a torus $\mathbb{T}^n$ (see for example ...
- 193
8
votes
1
answer
460
views
First order estimates of geodesic normal coordinates
Let $(M^n,g)$ be a complete Riemannian manifold with $|Rm| \le 1$. Can we find two positive constants $C$ and $\epsilon$, depending only on $n$, such that under the normal coordinates $(g_{ij})$ with ...
- 2,535
8
votes
2
answers
2k
views
Kähler metrics for projective space that are not the Fubini-Study metric
For projective $N$-space $CP^{N}$, there is a canonical Kähler metric called the Fubini-Study metric. Do there exist other Kähler metrics for $CP^N$. If so, is there any classification of such metrics?...
- 1,479
8
votes
0
answers
295
views
Intuition for the volume form - combinatorial definition?
I apologize that this is short of research level but I have realized that I am not happy with my understanding of the volume form on an oriented Riemannian manifold and I was hoping to find some ...
- 1,075
8
votes
1
answer
599
views
Exact condition for smooth homogeneous to imply Riemannian homogeneous for compact manifolds
Let $ (M,g) $ be a homogeneous Riemannian manifold. That is, the isometry group $ Iso(M,g) $ acts transitively on $ M $. Let $ \pi_1(M) $ be the fundamental group of $ M $. Then $ \pi_1(M) $ has ...
- 4,081
8
votes
1
answer
916
views
geodesic 2-dimensional submanifolds of a Riemannian manifold [duplicate]
Possible Duplicate:
Must a surface obtained by exponentiating a plane in a tangent space of a Riemannian manifold be geodesically convex?
The one dimensional geodesic submanifolds of a given ...
- 11.1k
8
votes
1
answer
421
views
$C^k$ one-parameter family of metrics
Consider a smooth Riemannian manifold $M$ and a $C^k$ one-parameter family of Riemannian metrics $g_t$ on $M$. Here $k$ could be any integer, $k$ could be infinity, when the one-parameter family $g_t$ ...
- 1,407
8
votes
1
answer
1k
views
Spectrum of the Laplacian on p-forms on the sphere
In this paper the authors give an explicit description of the eigenforms and spectrum of the Laplacian acting on $p$-forms on the round sphere $S^n$, apparently citing an unpublished computation of ...
- 585
8
votes
2
answers
452
views
Geodesic sphere in the octonion projective plane
I am considering Laplacian eigenvalues of a geodesic sphere in the octonion projective plane and I would like to know the metric on a geodesic sphere.
Does the metric on a geodesic sphere in the ...
- 141
8
votes
2
answers
2k
views
Tweetable way to see that Willmore energy is Möbius invariant?
Consider a compact orientable Riemannian manifold $M$ (without boundary) isometrically immersed into $\mathbb{R}^3$. The Willmore energy of $M$ is the functional
$$\mathcal{W} = \int_M H^2 dA$$
...
- 3,059
7
votes
1
answer
162
views
Estimate of number of boundary components of a compact Riemannian 2-surface
Let $X$ be a compact smooth 2-dimensional Riemannian manifold with boundary. Assume that the Gauss curvature of $X$ is at least $-1$ and the diameter is at most $D$. Assume that near the boundary the ...
- 21.8k
7
votes
1
answer
922
views
Trace and divergence of the Bach tensor
Let $(M, g)$ be a compact Riemannian manifold of dimension $n \geq 4$.
In Besse's book, the Bach tensor is defined as the gradient of the functional
$$
SW(g) = \int_M |W(g)|_g^2 d\mu^g
$$
meaning that,...
- 1,263
7
votes
1
answer
502
views
Fundamental groups of compact manifolds with non-negative Ricci curvature.
I would like to find an appropriate reference for the following statement:
Statement. Let $M$ be a compact Riemannian manifold with non-negative Ricci curvature.
Then $\pi_1(M)$ is virtually abelian.
...
- 14.3k
7
votes
1
answer
554
views
Minimal distance spheres in complex projective spaces
My question has to do with distance spheres in $\mathbb CP^{n+1}$. I am interested in knowing what is the radius $r$ of a distance sphere $S(r)$ around a point that makes it a minimal submanifold of $\...
- 5,891
7
votes
1
answer
368
views
Does complexified isometry group act transitively on tangent bundle of compact Riemannian manifold?
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\SU{SU}\DeclareMathOperator\O{O}\DeclareMathOperator\Iso{Iso}$Let $ g $ be the round metric on the sphere $ S^n $. Since $ S^...
- 4,081
7
votes
2
answers
2k
views
The integral of torsion
I found the following * exercise (exercise *9) in page 407 of the book of do Carmo "Differential geometry of curves and surfaces". This problem is a classical theorem which is referenced ...
- 356
7
votes
0
answers
352
views
Finitely generated projective modules over the algebra of sections of the Clifford bundle
Consider a (pseudo-)Riemannian manifold $(M,g)$ and the corresponding Clifford bundle $Cl_g(T^*M)$. Let $R$ be the algebra of sections of $CL_g(T^*M)$, with point-wise multiplication. What are the ...
- 21.5k
7
votes
1
answer
2k
views
Volume of geodesic balls
I have two questions (somewhat related) regarding local geometry on a SMOOTH, COMPACT Riemannian manifold. I still have a hard time getting a "good" understanding of local geometry.
Question 1:
It ...
- 207
7
votes
1
answer
841
views
Hilbert 16th problem via hyperbolic geometry
More than 16 years ago, I heard from someone that he thinks that there is a possible relation between Hilbert's 16th problem(for $n=2$) and Hyperbolic geometry. He says that a ...
- 356
7
votes
3
answers
2k
views
Changing coordinates so that one Riemannian metric matches another, up to second derivatives
Let $g$ and $g'$ be two $C^2$-smooth Riemannian metrics defined on neighborhoods $U$ and $U'$ of $0$ in $\mathbb R^2$, respectively. Suppose furthermore that the scalar curvature at the origin is $K$ ...
- 8,512
7
votes
1
answer
415
views
A diffeomorphism of the torus with constant singular values
Let $\mathbb{T}^2=\mathbb{S}^1 \times \mathbb{S}^1$ be the flat $2$-dimensional torus, and let $0<\sigma_1 < \sigma_2$ satisfy $\sigma_1 \sigma_2=1$.
Does there exist an area-preserving ...
- 6,741
7
votes
1
answer
558
views
minimal surfaces in $S^n$
Thanks to Choi-Schoen theorem, we know that the space of embedded minimal surfaces into $S^3$ of fixed genus is compact. My question are simples:
Can we remove the embeddness assumption?
Can we ...
- 914
7
votes
2
answers
499
views
Submanifolds of Lie groups with abelian normal bundle
Let $M$ be a submanifold of a symmetric space $Q$. The normal bundle $NM$ is called abelian if $\exp(N_{p}M)$ is contained in some totally geodesic and flat submanifold of $Q$ for all $p \in M$; see ...
- 965
7
votes
1
answer
502
views
Initially horizontal geodesic is always horizontal
I am trying to prove the following. (I posted this on math.se with no success)
Let $E,B$ be Riemannian manifolds. Suppose
$\pi: E\to B$ is a Riemannian submersion.
For each $x\in E$, define $V_x E = \...
- 305
7
votes
1
answer
2k
views
existence of totally geodesic hypersurfaces
Assume we are on a smooth, complete Riemannian manifold $(M,g), dim(M) \geq 3$. What are the specific geometric/topological constraints for such a manifold to admit complete, totally geodesic ...
- 133
6
votes
0
answers
260
views
Can a simple Riemannian metric on the disc be extended to a Zoll metric on the sphere?
Given a simple Riemannian metric $(D,g)$ on the two-disc---its geodesics have no conjugate point and the boundary of the disc is strictly convex---, is it possible to embed $(D,g)$ isometrically into ...
- 13.5k
6
votes
2
answers
2k
views
Poincare-like inequality on compact Riemannian manifolds
I am looking for a Poincare Inequality on balls but instead of euclidean space, I have a compact Riemannian manifold without boundary. The inequality I am looking for is the equivalent of
$$ \int_{...
- 63
6
votes
2
answers
903
views
Ricci curvature of the symplectic group
Is the Ricci curvature of the compact symplectic group $Sp(n)$ bounded below by $cn$ for some constant $c > 0$ independent of $n$?
For $O(n)$ and $U(n)$ I know many references which state such a ...
- 11.4k
6
votes
3
answers
833
views
Hypersurfaces and Elliptic Points
I'm reading a paper, in which we have $M^n$ an n-dimensional compact hypersurface embedded in $\mathbb{R}^{n+1}$. We take the scalar cuvature $R$ to be the elementary symmetric polynomial of degree 2 ...
- 1,123
6
votes
2
answers
808
views
Examples of two-dimensional Riemannian manifolds that can't be isometrically embedded into $\mathbb{R}^4$
Can anyone give some examples of two-dimensional Riemannian manifolds $(M,g)$ that can't be isometrically embedded into $\mathbb{R}^4$? (Further more Globally)
What if it is smooth?
- 273
6
votes
0
answers
537
views
Counting limit cycles via curvature in Riemannian geometry
In this post we would like to give a possible new approach to the second part of the Hilbert 16th problem
First we give a short introduction:
A quadratic system is a polynomial vector field on ...
- 356