All Questions
Tagged with reference-request dg.differential-geometry
800 questions
4
votes
3
answers
2k
views
Existence of a partition of unity with uniformly bounded derivatives.
Hi,
this is again a question from me which did not get any answer at math.stackexchange (Link: https://math.stackexchange.com/questions/27366/)
This question is about how well one can choose a ...
- 2,998
4
votes
4
answers
589
views
Measures of the complexity of a metric
I am seeking a measure of the "complexity" of a surface $S$,
a quantity that reflects how widely the metric varies from spot to
spot. I am primarily interested in surfaces topologically
equivalent to ...
- 151k
4
votes
3
answers
5k
views
Green's function on sphere
Consider radial (normal) coordinates on a sphere $S^n, n \geq 2$. Let the "origin" be the north pole $(0, 0,..., 1)$ and the coordinates be denoted by $(r, \theta)$. We know that the Laplacian $\...
- 41
4
votes
2
answers
347
views
good reference on brieskorn manifold
I am trying to learn something on the Brieskorn manifold (interested in the topological property)
Can the Mathoverflow Experts give me some good refencece (in English)?
By the way,is there an ...
- 153
4
votes
1
answer
486
views
Every _______ $d$-manifold has an $S$-structure
I am looking for some analogous nontrivial but known statements and references about statements of the form:
Every _______ $d$-manifold has an $S$-structure.
Here _______ is a placeholder for ...
- 10.5k
4
votes
1
answer
150
views
Linearisation of complex $S^1$ actions at fixed points
Let $(U,x)$ be an open complex $n$-manifold (say an $n$-ball) with an action of $S^1$ by holomorphic transformations that fix $x$. How to prove that there is a neighbourhood $U_1\subset U$ of $x$ ...
- 14.3k
4
votes
1
answer
1k
views
Christodoulou's paper on naked singularities in inhomogeneous dust collapse
I have been studying of late about formation of naked singularities in certain collapse scenarios in Einstein's theory. It seems to me that the canonical paper to read about how such a formation is ...
- 3,541
4
votes
2
answers
219
views
Is $\mathbb{P}T^*M$ a sub-Riemannian manifold if $M$ is Riemannian?
(this question is about a particular aspect of a previous question, which was not duly stressed)
Let $(M,g)$ a Riemannian $n$-dimensional manifold, and let
$$
\widetilde{M}:=\mathbb{P}T^*M
$$
be the $...
- 2,527
4
votes
1
answer
220
views
Are all bidimensional second-order PDE at most quadratic in the top derivatives of Monge-Ampère type?
The general Monge-Ampère equation in $n$ independent variables is a quasi-linear combination of all the possible minors of the $n\times n$ Hessian matrix
$$
\left\|\frac{\partial^2u}{\partial x^i\...
- 2,527
4
votes
2
answers
505
views
Eigenfunctions of the Laplacian on singular spaces
Consider a compact manifold $M$ with boundary and corner. As an example, we could have the cube $\{(x_1, x_2,..x_n) \in \mathbb{R}^n : x_i \in [0,1]\}$. We could very well define the Laplacian $\Delta$...
- 43
4
votes
2
answers
375
views
Converse to Chow's theorem in sub-riemannian geometry
Chow's theorem is the statement that if $M$ is a connected smooth manifold endowed with a distribution $\mathcal{D}$ which is completely non integrable (i.e. iterated commutators of smooth sections of ...
- 2,479
4
votes
1
answer
129
views
Holomorphic Poisson structures on $C P^{n-1}$ and homogeneous Poisson structures on $C^n$
Is it correct that any holomorphic Poisson structure on $C P^{n-1}$ can be lifted to a homogeneous Poisson structure on $C^n$? By homogeneous I mean a quadratic Poisson structure of the form $\{z_i,...
4
votes
1
answer
699
views
Spectrum of the Laplace-Beltrami operator on $L^p$: where is it?
On a noncompact Riemannian manifold $M$, the $L^2$-spectrum of the Laplace-Beltrami operator $\Delta$ sits inside $\mathbb{R}$ (by self-adjointness), either to the left or to the right of $0$ ...
user14166
4
votes
2
answers
2k
views
An introduction paper or book to Spectral Flow
Is there someone can tell me some papers or books about the basic material of Spectral flow? I want to know, what is spectral flow and how to use it to geometry.
- 381
4
votes
1
answer
198
views
Examples of hyperbolic manifolds of dimension $\geq$ 3 with disjoint totally geodesic hypersurfaces
I am hoping to find examples of compact hyperbolic manifolds with at least 2 disjoint totally geodesic hypersurfaces. Ideally, I would like examples in dimension at least 4, though 3-dimensional ...
- 6,025
4
votes
1
answer
295
views
On a result of Cartan for homogeneous manifolds arising from a quotient of discrete subgroups
I'm not sure if this is completely relevant to MO, let me know if this would be better on MSE.
I have been told today by a professor of mine that the following is a classic result of Cartan. Suppose $...
- 1,763
4
votes
1
answer
728
views
Simplicity of the first Laplace-Beltrami eigenvalue on Riemannian manifolds
On a compact Riemannian manifold $M$ (we assume Dirichlet boundary condition if $\partial M \neq \emptyset$), the Laplace-Beltrami operator $-\Delta$ has a discrete spectrum $0 < \lambda_1 \leq \...
- 43
4
votes
2
answers
758
views
Riemannian metric of hyperbolic plane
I'm fishing for the origin of the idea to consider "trace scalar product" on the space of ($G$-)orthogonal projectors as means of defining a Riemannian metric on some subset of lines in a vector space....
- 8,597
4
votes
1
answer
733
views
Reference request: Intrinsic definition of the strong Whitney topology on $\mathcal{C}^{\infty}(M,\mathbb{R})$ without using charts or jets
Let $M$ and $N$ be smooth manifolds. There is a description of the strong Whitney topology on $\mathcal{C}^{\infty}(M,N)$ in terms of partial derivative in charts (using locally finite sets of charts ...
- 191
4
votes
2
answers
475
views
How to compute the index of such operator?
Let $M$ be a compact Riemannian manifold, with $R$ nowhere-vanishing vector field on $M$(whose orbit may be closed/ not closed). $E$ and $F$ are two vector bundle (Edit: which are sub-bundles of $\...
- 857
4
votes
1
answer
434
views
Curvature and Symmetry on Kähler manifolds
Hi there,
Suppose $X$ is a Kähler manifold that has an analytic isometry $S$, with $S^k = \operatorname{Id}$ ($k \in \Bbb N$). In a situation like this (maybe with additional assumptions on $X$) can ...
- 1,211
4
votes
1
answer
334
views
Non-commutative versions of X/G
Let $X$ be a Riemannian manifold and let $G$ be a (at most countable, if that matters) discrete group acting properly and by isometries on $X$. Let $\mathcal{O}$ be the sheaf of analytic functions on ...
4
votes
1
answer
194
views
Chern numbers of almost complex manifolds
Suppose we are given two integer numbers $p$ and $q$ such that $p+q\equiv 0 \pmod{12}$. There is a result saying that for every such pair there exists a non necessarily connected almost complex ...
- 2,305
4
votes
3
answers
669
views
Gaussian bounds on Dirichlet heat kernel
Let $(M, g)$ be a compact Riemannian manifold and let $p(t, x , y)$ be the heat kernel of $M$. Then there exist constants $c, C > 0
$ such that $$\frac{c}{t^{n/2}}
e^{-\frac{1}{4t}d(x, y)^2} \leq ...
- 81
4
votes
2
answers
281
views
Heat kernel asymptotics for small distances
I heard a talk where the speaker said that on a Riemannian manifold, for small values of $\text{dist }(x, y)$, the heat kernel $p_t(x, y)$ satisfies
$$p_t(x, y) = \frac{1}{(4\pi t)^{n/2}}e^{-\frac{\...
- 41
4
votes
1
answer
710
views
Reference request for instantons
I've been researching instantons lately and I'd like to learn more about them but would like some help finding what to read. I have read about the ADHM equations and their noncommutative analogues. ...
- 161
4
votes
1
answer
478
views
Geometry of ends of a finite volume negatively curved manifold
Is there a survey of the geometry of manifolds with finite volume Riemannian metrics of negative sectional curvature? More specifically, I am interested in the geometry of cusp ends of such manifolds, ...
- 43
4
votes
1
answer
646
views
Combinatorial geodesics
[There has been a flaw in my definition - as Sergei and Andreas pointed out. I hope I could fix it.]
I want to understand how the concepts of directions, straight (or shortest) lines, and geodesics &...
- 9,720
4
votes
1
answer
479
views
Work on an Einstein-Hilbert type action but with the *absolute value* of scalar curvature?
This is only my second question on mathoverflow, so my apologies if this would be more appropriate at a physics site. My question concerns a modification to the Einstein-Hilbert action. The standard ...
- 175
4
votes
2
answers
1k
views
Special Killing Vector Fields
Consider $(M^{n},g)$ to be a Riemannian manifold and suppose that $X$ is a smooth non-trivial Killing vector field on $M$. Away from the zeros of $X$ we have a natural distribution $D$ of $(n-1)$-...
- 2,299
4
votes
1
answer
172
views
Viscosity solutions of eikonal equation on Riemannian manifolds
It is well known that given a bounded open region $\Omega \subset \mathbb{R}^n$, the Dirichlet problem $$\lVert \nabla u \rVert = 1, \quad u|_{\partial \Omega} = 0$$
admits the unique viscosity ...
- 235
4
votes
2
answers
405
views
Gaussian upper heat kernel bounds on closed Riemannian manifolds
Let $M$ be a closed Riemannian manifold, and let $h(t, x, y)$ denote the heat kernel on $M$. We know that there exists short time upper Gaussian heat kernel bounds of the following kind:
$$
h(t, x, y) ...
- 1,407
4
votes
1
answer
139
views
Convex hull of a connected subset on a complete surface of non-positive curvature
Let $S$ be a simply connected surface, possibly with boundary components, with a smooth complete metric of non-positive curvature. Let $X\subset S$ be a closed connected subset. I would like to know ...
- 14.3k
4
votes
1
answer
416
views
Some notational questions regarding tangent vectors
I am not a specialist in differential geometry, so I have some difficulties in finding the right words for the following natural things:
First of all it seems that there is a lot of nonequivalent ...
- 5,529
4
votes
1
answer
532
views
Diffusion semigroup generated by Laplacian
Let $M$ be a complete Riemannian manifold and $\Delta$ denote the Laplacian on it. Also assume that the spectrum of $-\Delta$ lies inside $[a, \infty)$. Let $P_t, t > 0$ denote the diffusion ...
- 43
4
votes
1
answer
503
views
singular metric (with essential singularity)
Working on some $Q$-curvature equation in dimension $4$, I have been faced with singular metric of the form $(\mathbb{B}, e^{-1/\vert x\vert ^2} \vert dx\vert)$. I try to figure out to what those ...
- 914
4
votes
1
answer
134
views
Name for Curves from Driving on Smooth Manifolds
Is there already name for the generalization of Clothoids to curves on smooth manifolds, i.e. where the curve's curvature depends linearly on the curve's length-parameter?
In the euclidean plane ...
- 13.2k
4
votes
1
answer
565
views
Riccati equation and principal curvatures
Let $\Omega$ be an open subset of a Riemannian manifold $M$. Assume that $\Sigma:=\partial \Omega$ is $C^2$.
Let $U$ be a neighborhood of $\Omega$ such that $\exp_p(t\nu(p))$ is diffieomorphism, ...
- 143
4
votes
1
answer
304
views
Local product structure of determinantal variety
The variety $X_n$ of singular $n\times n$ real matrices is stratified by smooth strata $X_{n,k}$ where $k$ is the rank. Choose a rank $k$ matrix $A\in X_{n,k}$. Is there a local diffeomorphism sending ...
- 16.6k
4
votes
1
answer
301
views
Injectivity radius of parallel hypersurfaces
Let $(M,g)$ be a Riemannian manifold and let $N$ be a compact hypersurface isometrically embedded into $M$ and let $\eta$ denote a choice of unit normal vector field on $N$. It is then true that $N$ ...
- 409
4
votes
1
answer
317
views
Minimal graph over convex domain is area-minimizing
I am looking for a reference stating that
If a graph $z=f(x,y)$ over a convex domain $D$ is minimal, then it is area-minimizing.
5.4.18 in Federer's "Geometric measure theory" and Lemma 1.1. in ...
- 45k
4
votes
1
answer
347
views
Some questions on a paper of Wilking
I am currently trying to understand Wilking's paper "A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities" (DOI: 10.1515/crelle.2012.018, arXiv:1011....
- 81
4
votes
1
answer
215
views
Does a $G$-structure on $M$ automatically descend to a contact $G$-structure on $\mathbb{P}T^*M$?
If $M$ is a real smooth manifold of dimension $n+1$, by $D\in\mathbb{P}T^*M$ I mean a tangent hyperplane at some point of $M$. I denote by $b$ the canonical projection of the $(2n+1)$-dimensional ...
- 2,527
4
votes
0
answers
179
views
Recognize this metric? Do you have a name for this metric on the product of spheres?
Take the product $S^2 \times S^2$ of two two-spheres,
but perturb the product metric as follows.
Think of each $S^2$ as the unit two-sphere in Euclidean 3-space
in the standard way
so that for $p ...
- 8,336
4
votes
0
answers
128
views
Errata for "Foliations and Geometric Structures" by Aurel Bejancu and Hani Reda Farran
I'm reading "Foliations and Geometric Structures" (2006) by Aurel Bejancu and Hani Reda Farran and have been looking for an errata sheet. Unfortunately Prof. Bejancu has passed away. I ...
4
votes
0
answers
167
views
What textbooks/papers should I read to try to make this rigorous?
Consider a surface of revolution $S$ and an embedding $e:S \hookrightarrow X^3$ for $X^3=[0,1]^3$ with cone points $p,q$ elements of $\partial X^3$ where $\partial X^3=X^3-(0,1)^3$ for $\mathrm {sup}~ ...
- 383
4
votes
0
answers
116
views
Do any Legendrian knots in standard contact 3-space have big tubular neighborhoods?
Consider $\mathbb{R}^3$ with the standard contact structure $\ker(dz-y\,dx)$.
According to the contact version of Weinstein's theorem, any Legendrian knot $L\subset \mathbb{R}^3$ has a tubular ...
- 501
4
votes
0
answers
72
views
Riemannian manifolds with a unique distance property
Let $M$ be a compact Riemannian manifold with geodesic distance function $d$, of (normalised) diameter $1$.
Some of my favourite manifolds $M$ have the property that there exists an integer $k$ such ...
- 1,949
4
votes
0
answers
114
views
Degeneration formula and Donaldson-Floer theory
Is there a relation between the degeneration formula of GW Invariants of Jun Li and the Donaldson-Floer theory? Is there an example / discussion anywhere of/on this relation?
- 153
4
votes
0
answers
81
views
Pseudometrics on world lines
Consider the space $W$ of smooth time-like curves in $\mathbb{R}^{n,1}$ with fixed ends.
Given $\gamma\in W$, consider the space $T_\gamma$ of all smooth normal fields along $\gamma$;
one may think ...
- 14.3k