All Questions
Tagged with dg.differential-geometry riemannian-geometry
1,985 questions
1
vote
2
answers
290
views
Examples on small cut radius of totally convex set in non-negatively curved manifold
Suppose $M^n$ is an open complete nonnegatively curved Riemannian manifold. In Cheeger-Gromoll's proof of the soul theorem. They need an estimate on the cut radius of a totally convex set $C$. By a ...
- 283
12
votes
1
answer
720
views
Is a manifold with flat ends of bounded geometry?
A Riemannian manifold $(M,g)$ is said to have flat ends if the curvature tensor of $g$ vanishes outside a compact set $K$. I was wondering if such manifolds are of bounded geometry. Recall that a ...
- 7,583
7
votes
2
answers
807
views
Asymptotic expansion of the Schrödinger kernel?
My stackexchange post was somewhat unsatisfactory (also because I may not have stated clear enough what my interest was). So here it goes!
Let $M$ be a compact Riemannian manifold and $\Delta$ be the ...
- 9,853
4
votes
2
answers
626
views
Uniqueness of Kähler form with same volume
Hallo,
Let $M$ be a compact real-analytic Riemannian manifold with Riemannian metric $g$. Let $U \subset T^{*}M$ be a open neighbourhood of the zero section. On $U$ there exists a complex structure $...
- 339
3
votes
0
answers
242
views
What is known about analogous results of Kazdan and Warner in higher dimensions?
First let me state a Theorem due to Kazdan and Warner:
``Let M be a compact two dimensional orientable manifold. Let
$f: M \rightarrow \mathbb{R}$ be a function that has the same
sign as $\chi(M)$,...
- 3,245
26
votes
5
answers
7k
views
Intuition for mean curvature
I would like to get some intuitive feeling for the mean curvature. The mean curvature of a hypersurface in a Riemannian manifold by definition is the trace of the second fundamental form.
Is there ...
- 693
9
votes
3
answers
479
views
twisted Poisson structures, degenerate metrics and integrability properties of (2,0)-tensors
Given a regular (constant rank) bi-vector $\Pi \in \Gamma(\bigwedge^2TM)$ on a smooth manifold $M$ the necessary and sufficient condition for the image of $\Pi^\sharp:T^*M\to TM$ to be an integrable ...
- 654
6
votes
2
answers
633
views
Rigorous solution to Ricci Flow on dumbbell $S^3$
To begin a small interest in Ricci Flow and similar tools, I am starting with Hamilton's expository paper The Formation of Singularities in the Ricci Flow. This was posted in 1995, so I am wondering ...
- 17.5k
1
vote
1
answer
218
views
Isometric embedding of a neighbourhood of a totally real submanifold in a Kähler manifold
Hallo,
Let $(M,J,\omega)$ be a real-analytic Kähler manifold. Let furthermore $A \subset M$ be a real analytic, totally real, Lagrangian submanifold and set $g := h|_{A}$. Where $h$ is the Kähler ...
- 339
8
votes
1
answer
458
views
Different complexifications of a real analytic Riemannian manifold
I have a question concerning the complexification of a real analytic Riemannian manifold. Let $(M,g)$ be a compact Riemannian manifold. It is a well known fact that in a neighbourhood $U$ of the zero ...
- 101
3
votes
2
answers
669
views
Real analytic submanifolds of $\mathbb{R}^{n}$
Hallo,
Let $(M,g)$ be a Riemannian $k$-dim real analytic submanifold of $\mathbb{R}^{n}$. Is it true that $M$ in $\mathbb{R}^{n}$ looks locally (in a small neigbourhood around some point in $M$) as ...
- 339
2
votes
1
answer
283
views
Isometric embedding of a compact Lie Group in $M(n,\mathbb{C})$
Greetings,
Let $G$ be a compact Lie group with a bi-invariant inner product $h$ on it. Can one embedd $G$ in $M(n,\mathbb{C})$ isometrically for some $n \in \mathbb{N}$. By isometrically I mean that ...
- 101
3
votes
1
answer
388
views
Isometric embedding of a real-analytic Riemannian manifold in a compact Kähler manifold
Hallo,
It is a known fact that any real-analytic Riemannian manifold $M$ admits a isometric embedding in a Kähler manifold $\Omega$, where $M$ is totally real in $\Omega$. Of $\Omega$ can be taught ...
- 339
15
votes
0
answers
637
views
"Homogeneity" of the Hopf fibration $S^7\to S^{15}\to S^8$ [closed]
My question has to do with an apparent contradiction I get regarding the Hopf fibration $S^7\to S^{15}\to S^8$. Namely, the two following statements cannot be true at the same time (but I do not see ...
- 5,891
5
votes
1
answer
339
views
Are there countably many diffeomorphism classes of finite radius balls of complete Riemannian manifolds?
Suppose $M$ is a smooth complete Riemannian manifold and $x$ is a point in $M$. For any positive radius $r$ we consider the open ball $B(x,r)$ centered at $x$ with radius $r$.
If we ignore the ...
- 4,304
2
votes
0
answers
179
views
About Thom Theorem (representation submanifold for $H_{n-2}(M^n)$)
Recall Thom theorem : If $M^n$ is a smooth orientable closed manifold then any homology class in $H_{n-2}(M)$ is represented by the fundamental class of a smooth submanifold.
And in the Harper and ...
- 1,100
13
votes
2
answers
3k
views
Intuition for Levi-Civita connection?
Levi-Civita connection is usually defined as the unique connection which is torsion free and preserves metric.
Question Is there some intuitively transparent constructive way to define it (or ...
- 24.9k
7
votes
1
answer
332
views
volume of exceptional group orbits
Assume that $G$ is a compact group acting by isometries on a (compact) Riemannian manifold (M,g), with principal orbits of dimension $d>0$. For $x\in M$, let $G(x)$ denote the $G$-orbit of $x$, by $...
0
votes
1
answer
339
views
Polarisation in a neighbourhood of a Lagrangian submanifold
Let $(X, \omega)$ be a symplectic manifold of dimension $2n$ and $\omega$ is an exact symplectic form i.e. $\omega = -d\alpha$. Let furthermore $M \subset X$ be a compact Lagrangian submanifold such ...
- 339
3
votes
3
answers
1k
views
Rotation in Hyperkähler manifolds
Any Hyperkähler manifold has 3 complex structures $I_{1}, I_{2}, I_{3}$. Assume that there is an additional complex structure $J$. Can this be written as $J = aI_{1} + bI_{2} + cI_{3}$, where $(a,b,c) ...
- 339
4
votes
2
answers
575
views
Do transvers foliations induce complex structure?
Hallo,
I have the following question: Let $M$ smooth analytic manifold of dimension 4n. Assume furthermore that $M$ admits two foliations $A$, $B$, both with leaves of dimension 2n such that the ...
- 41
3
votes
0
answers
585
views
Differentiation of Logarithm Map in Riemannian Geometry
I have a simple question regarding the differentiation of the logarithm mapping in Riemannian manifolds:
Assume that $M$ is a compact Riemannian manifold, isometrically embedded into $\mathbb{R}^n$.
...
- 233
14
votes
2
answers
964
views
A riemannian manifold with finitely many closed contractible geodesics
By a closed geodesic, I mean a smooth periodic geodesic $\mathbb{R} \rightarrow (M,g)$. I will consider them up to geometric distinction.
This means that any two closed geodesics are equivalent if ...
- 847
6
votes
1
answer
1k
views
Holonomy of a Kähler manifold
Hi,
I have the following question: Let $(M,J, \omega)$ be a Kähler manifold (not necessary compact). We know that the holonomy group is a subgroup of $U_{n}$. Let $\Omega$ be a constant ($\nabla \...
- 93
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
4
votes
1
answer
1k
views
Curvature as metric invariant
This is quite well-known: the ONLY metric invariants are curvature, its higher
derivatives, and any possible contractions between them.
The meaning of an invariant is, to put it simply, a tensor ...
- 133
7
votes
1
answer
496
views
Open problems about CMC hypersurfaces with symmetries?
Recently, Andrews and Li announced a complete classification of CMC ($H=const.$) tori in $S^3$, confirming a conjecture of Pinkall and Sterling. Their main result is that any such torus is ...
- 5,891
8
votes
0
answers
422
views
Exhaustion of an open manifold of bounded curvature and finite volume
In the Cheeger-Gromov paper "On the Characteristic Numbers of Complete Manifolds of Bounded Curvature and Finite Volume",
http://www.maths.ed.ac.uk/~aar/papers/cheegergr1.pdf,
the authors make the ...
- 2,623
3
votes
1
answer
745
views
Star-shaped domain in a space form
Let $M$ be either $\mathbb R^n$, $\mathbb H^n$ or $\mathbb S^n$ and $p\in M$, by a star-shaped domain w.r.t $p$ I mean a connected open subset $\Omega$ in $M$ containing $p$ such that its boundary is ...
- 103
13
votes
3
answers
851
views
Are negatively pinched manifold locally conformally flat?
One knows that hyperbolic manifolds are locally conformally flat.
How about those negatively pinched manifolds, i.e. the sectional curvature $K$ satisfy:
$$
-\Lambda \le K \le -\lambda$$
for $\Lambda&...
- 2,623
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
8
votes
2
answers
2k
views
Almost constant bump function
I ran into the following situation and it turned out to be more subtle than it looked.
I have a complete Riemannian manifold $M$ and my objective is to construct a sequence of functions $f:M \to [0,1]...
- 4,304
2
votes
1
answer
807
views
a result of soul theorem,right?
$X$ is an $n$-dim positively curved manifold and $Y$ is a totally geodesic submanifold of codimension 1. Then cutting along $Y$ we get $n$-dim positively curved manifolds without boundary, by soul ...
- 689
3
votes
1
answer
5k
views
Relationship between sectional curvature, bisectional curvature and conjugate points
Tian has defined bisectional curvature for unit and perpendicular tangent vectors $X,Y$ as follow
$$R(X,Y,X,Y)+R(X,JY,X,JY).$$
If bisectional curvature be constant, is there any relationship between ...
- 105
7
votes
3
answers
736
views
Nearly constant curvature implies "nearly isometric" to a space form?
It is well known a Riemannian manifold with constant sectional curvature is a quotient of the Euclidean space, hyperbolic space or sphere. In particular we know how their metric looks like locally.
...
- 103
2
votes
1
answer
484
views
Sobolev norm of distance function on Riemannian manifold
$\DeclareMathOperator\SL{SL}$Suppose $M$ is a Riemannian manifold with distance function $d:M\times M \rightarrow [0,\infty)$. If it helps let $M$ be a Lie group with finite Haar measure $\mu$ and ...
- 23
1
vote
1
answer
423
views
Extension of groups in Bieberbach's theorem
I am reading de la Harpe's book "Topics in Geometric Group Theory".
On page 145, there is a theorem:
Let $V$ be a complete $n$-Riemannian manifold with sectional curvature satisfying $K\ge 0$. Then ...
- 1,101
2
votes
1
answer
425
views
holomorphic extension of forms
hallo,
I have the following question: Let $M$ be a $n-$dimensional complex manifold and $X \subset M$ be a compact $n-$dimensional totally real analytic Riemannian submanifold. Let furthermore $\...
- 29
9
votes
2
answers
638
views
Curvature of the Cayley projective plane
The Cayley projective plane can be realized as the compact homogeneous space $F_4/\mathrm{Spin}(9)$. In this way one can compute the curvature of this symmetric space in terms of a suitable ...
- 403
19
votes
1
answer
2k
views
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 ...
- 7,831
4
votes
1
answer
201
views
estimate over simply-connected Riemannian manifold with non-positive sectional curvature
Let $M$ be a Complete simply-connected $n$-dimensional Riemannian manifold with non-positive curvature,$\Omega $ is a open subset of $M$ ,If $n\geq 4$ Is anyone give a estimate of $\frac{Vol_{n}\...
- 119
9
votes
2
answers
367
views
Is compact flat manifold cusp cross-sections of a complete finite volume hyperbolic manifold?
Let $M^{n-1}$ be a closed flat manifold. Is it true that there exists a hyperbolic manifold $N^n$ with finite volume such that $M$ is a cusp cross-section of $N$?
It was proved in "On the geometric ...
- 2,623
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
4
votes
1
answer
2k
views
Totally geodesic submanifold of round sphere
Let $S^n$ be the $n$-dimensional round sphere (i.e. with Riemannian metric of constant curvature +1). Is there any classification result of totally geodesic embedded submanifolds? Are they all round ...
- 2,623
0
votes
1
answer
314
views
G-structures and complete riemannian manifolds
what are possible fundamental and introductory texts about G-structures ?
and where i can find the proof of this proposition:
if G(group) acts properly discontinuously on a space X , then G is a ...
- 165
9
votes
1
answer
2k
views
Isoperimetry and Poincaré Inequality
What are the known relations between isoperimetric and Poincaré inequalities on manifolds?
For example, for manifolds with a lower bound on Ricci curvature, the Cheeger-Buser inequality relates the ...
- 10.4k
10
votes
1
answer
820
views
A strange question about closed geodesics on a closed manifold
I'm studying a particular kind of curve evolution on Riemannian manifolds. It would help me
to know the answer to the following kinda weird question:
Does there exist a closed Riemannian manifold $M$ ...
- 545
2
votes
1
answer
551
views
Heisenberg group: research themes
I am currently studying the Heisenberg group from the Riemannian geometry point of view, particularly focusing on its Gromov boundary and more generally its metric properties.
I would like to know ...
- 31
3
votes
2
answers
845
views
How to solve the $C^\alpha$ Poisson equation on closed Riemannian manifolds?
To be specific, suppose $M$ is a closed oriented manifold, $g$ is a Riemannian metric of $M$.
Let $\Delta_g$ be the Laplace-Beltrami operator w.r.t. $g$.
Prove: Suppose $f\in C^\alpha(M)$ satisfies ...
- 1,083
5
votes
2
answers
1k
views
Kähler potentials that depend only on geodesic distance
Hermitian symmetric spaces of constant curvature have the property that the potential for their Kähler metric can be expresed as some function of the geodesic distance. Does anyone know if there are ...
- 1,378