All Questions
Tagged with reference-request dg.differential-geometry
800 questions
14
votes
3
answers
2k
views
Is there a "unique" homogeneous contact structure on odd-dimensional spheres?
Let $S^{2n-1}\subset\mathbb{C}^{n}$, and denote by $\langle\,\cdot\,,\,\cdot\,\rangle$ the Hermitian product. Then
$$
\mathcal{C}_p:=\{\xi\in T_pS^{2n-1}\mid\langle p,\xi\rangle=0\},\quad p\in S^{2n-1}...
- 2,527
10
votes
0
answers
415
views
Singularities in Yang Mills Flow
In "The Yang Mills flow in four dimensions", M. Struwe proves that this flow converges, up to bubbling phenomena. And he has conjectured that this explosion in finite time should happen as proven for ...
- 914
5
votes
1
answer
442
views
Chern-Weil theory for degenerated metric
If $\omega$ is a Kähler metric on a compact complex manifold $X$, the standard Chern-Weil theory says that the Chern classes $c_{i}(M)$ can be represented by forms involving the curvature of $\omega$....
- 51
1
vote
0
answers
81
views
Ring structure for $K^{-1}$?
My questions are
whether there exists a product structure for $K^{-1}(X)$? Here $K^{-1}$ is the odd topological $K$-group, and $X$ is a compact space (or a manifold), say.
If such a ring structure ...
- 1,173
3
votes
0
answers
159
views
Strictly Convex Smoothing of a function defined on an affine manifold
A function defined on an interval is strongly convex if it has positive definite second derivative. A function on an affine manifold is strongly convex
if its restriction to each line is. An affine ...
- 700
5
votes
2
answers
821
views
Is the Lie quadric $Q^3$ isomorphic to the Lagrangian Grassmannian $\operatorname{LG}(2,4)$?
$\DeclareMathOperator\LG{LG}$In the paper The - Conformal geometry of surfaces in the Lagrangian—Grassmannian and second order PDE (published on Proc. London Math. Soc.), I've found an interesting ...
- 2,527
2
votes
0
answers
282
views
Generalizing a result of Paul Andi Nagy
I am trying to generalizate a result of Paul Andi Nagy which says that in a almost Kähler manifold with parallel torsion we have $\langle \rho,\Phi-\Psi\rangle $ is a nonnegative number; in fact
$$
4\...
- 21
7
votes
1
answer
420
views
How does the kernel of the map $\Omega^{\bullet}(X)\rightarrow \Omega^{\bullet}(G\times X)$ relate to equivariant cohomology?
This question may be trivial for experts. Consider a (compact, connected) smooth manifold $X$ and a (compact connected) Lie group $G$ act on $X$. Then we have the action map
$$
\mu: G\times X\...
- 9,019
3
votes
1
answer
228
views
Exposition of the Calabi complex
I am interested in a complex derived by Eugenio Calabi in his article "On compact Riemannian manifolds with constant curvature".
The complex is referenced as "Calabi complex" in various citing ...
- 5,327
0
votes
0
answers
289
views
Third variation of area of a minimal surface
There is a formula for the third variation of area on page 96 of Nitsche's book,
Lectures on Minimal Surfaces, vol. 1 (English version). He says at the bottom of the
page it is good for normal ...
- 1,233
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
9
votes
1
answer
509
views
A question on generalized Einstein metrics on four-dimensional manifolds
I am thinking of a possible generalization of Einstein metrics (or a possible characterization of Einstein metrics) on four-dimensional manifolds,
\begin{equation*}
\mathrm{Ric}\circ\mathrm{Ric}=\...
- 399
5
votes
2
answers
704
views
Ricci curvature under rough convergence
From the work of Lott--Villani and Sturm, I know that the following fact holds:
(*) Suppose that $(M_k,g_k,dvol_{g_k})$ is a sequence of compact Riemannian manifolds of non-negative Ricci curvature ...
- 7,197
8
votes
1
answer
336
views
Short examples that are/are not quantum-ergodic
Are there any considerably short examples of manifolds that are/aren't quantum ergodic, or quantum unique ergodic?
Note that a (compact) Riemannian manifold is said to be quantum ergodic if almost ...
- 131
1
vote
0
answers
197
views
Euler class and self-intersection number of a surface in a 4-manifold [duplicate]
In the first two paragraphs of Circle bundles over $CP^1$ and self-intersection number of $CP^1$ embeddings, it is claimed that
For a compact oriented surface $X$ in a 4-dimensional oriented ...
- 655
5
votes
1
answer
906
views
Boundaries of relatively hyperbolic groups
When the interior of an n-manifold $M$ has a pinched negative curvature metric of finite volume, then its fundamental group $\Gamma=\pi_1M$ is relatively hyperbolic relative to the parabolic groups $\...
- 10.4k
0
votes
1
answer
153
views
Entropy of Negatively pinched manifolds
Suppose $M$ is a compact negatively pinched Riemannian manifold of dimension $n$. We normalize the metric such that $-1\le K\le -a^2$ for some $0<a\le 1$. Let $G$ be the fundamental group of $M$. ...
- 1,101
10
votes
1
answer
470
views
Monograph or rich survey on infinite-dimensional Riemann manifolds
I'm working with the space of smooth curves $\mathcal{C}$ in a smooth manifold $M$, having (different, pre-determined) fixed endpoints. I'd like to endow it with a Riemann structure (I already have a ...
- 5,407
3
votes
1
answer
525
views
Kahler-Einstein metrics on Toric manifolds are Torus-invariant?
let $(M,\omega)$ be a Kahler-Einstein toric manifold of complex dimension $m$. By toric manifold i mean a manifold that has an open dense subset $X$ biholomorphic to an algebraic torus $\mathbb{T}^{m}...
- 1,727
2
votes
1
answer
963
views
Lie-derivative of tensor field along tensor field
What is the natural notion of the Lie-derivative of a tensor field along another tensor field, and where can I find an exposition of that?
- 5,327
4
votes
0
answers
205
views
Where is the Courant operad discussed?
Where is the Courant operad discussed? And hopefully defined precisely.
By the Courant operad or rather a suitable generalization of operad to accommodate the inner product, the operad whose ...
- 3,880
11
votes
0
answers
501
views
The cones for Bochner–Lichnerowicz–Weitzenböck formula
The Bochner–Lichnerowicz–Weitzenböck formula can be written the following way
$$ \Delta \phi-\nabla^*\nabla \phi= R(\phi),$$
here $\phi$ is a section in a Dirac bundle and $R$ the something which can ...
- 1,785
8
votes
1
answer
633
views
Flag manifolds (=R-spaces): quotients by parabolic subgroups vs. isotropy representation
Real flag manifolds (also known as R-spaces) can be defined in two ways which I believe are equivalent although some fine print may have escaped me:
as a quotient of a semisimple real Lie group $G$ ...
- 32.5k
18
votes
2
answers
4k
views
Reference request: Geodesic flow on a manifold with negative curvature is ergodic
I'm reading about the Mostow's rigidity theorem, and the proof uses the following (maybe well-known) result:
The geodesic flow on a manifold with negative curvature is ergodic.
The lecture note that ...
- 957
1
vote
2
answers
534
views
Are there some websites for self learining of advanced mathematics? [closed]
Are there some websites for self learining of advanced mathematics?
For example, some great lecture vedio of differential geometry, Lie group , Lie algebra, algebraic topology and so on. Thanks
- 977
14
votes
1
answer
1k
views
Spectrum of Laplacian in non-compact manifolds
What can be said about the spectrum of the Laplace-Beltrami operator on a non-compact, complete Riemannian manifold of finite volume? For example, is the point spectrum non-empty?
What would be a ...
- 13.5k
2
votes
1
answer
255
views
Parameter dependent differential equation in a Lie group
It is well-known that a linear differential equation in a finite-dimensional vector space depends continuously on some external parameters (for details see below). I search for an explicit reference ...
- 5,824
39
votes
10
answers
4k
views
Are there some other notions of "curvature" which measure how space curves?
I am learning differential geometry and have a few questions on curvature. -- Background:
Gauss invented "Gauss curvature" to measure how surface curves.
Riemann gives an ingenious generalization of ...
0
votes
2
answers
435
views
Isomorphism of connections on a complex line bundle
Reading an article I faced with the following theorem, please give me a reference to a proof of the fact which is stated without any reference in the article. Is it a well-known fact?
Theorem. Let $E ...
- 1,329
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
2
votes
1
answer
490
views
Curve on a surface defined by its geodesic curvature
Suppose that $S$ is a smooth complete surface, and $c\colon [0,L]\to S$ is a smooth curve in $S$, parametrized by arc-length. Then $c$ is uniquely determined by its initial tangent vector and its ...
15
votes
3
answers
1k
views
moduli spaces are kahler?
I often heard from experts that "moduli spaces are Kahler". This sounds as a meta-theorem asserting that every time one defines reasonable moduli spaces, then there is a standard strategy to see (...
- 829
2
votes
0
answers
189
views
About the Lie algebra of polyvector fields
I would like to know if someone already did some computations of the group of Lie algebra automorphisms of the algebra of polyvector fields on $\mathbb{R}^n$ equiped with the Schouten bracket (or ...
- 1,609
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
1
vote
0
answers
463
views
Reference request for parallel transport
I am learning about parallel transport on a Riemannian manifold equipped with an affine connexion. It seems (if I understand it well) that, in general, we might not be able to compute the parallel ...
- 119
11
votes
2
answers
2k
views
Central extension of the algebraic loop group
I'm doing some constructions with the universal central extension $\widehat{\Omega G}$ of the loop group $\Omega G$ (here $G$ is a matrix group), where a priori the loops involved are just smooth, but ...
- 35.5k
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
1
vote
1
answer
178
views
Casimir of a three dimensional solvable lie algebra
Good morning everyone. I've encountered recently during my computations the following lie algebra
$$\mathfrak g=\text{span}(f_0,f_1,f_2),$$
with $$\begin{eqnarray}[f_2,f_1]&=&f_0+a f_2,\\ [...
user17697
6
votes
1
answer
753
views
Rigidity of secondary characteristic classes
For a representation $\rho:\pi_1M\rightarrow GL(n,C)$ and the associated flat $GL(n,C)$-bundle $E_\rho\rightarrow M$ one has the Cheeger-Chern-Simons classes
$$\hat{c}_k(E_\rho)\in H^{2k-1}(M,R/Z)$$
...
- 10.4k
2
votes
1
answer
391
views
Fully non-linear PDE
A nice method of obtaining existence of solutions of many geometrically defined (and hence highly degenerate) parabolic systems (such as mean curvature flow) involves the reduction of the system to a ...
2
votes
1
answer
338
views
Reference on Deligne-Mumford compactness for Riemann surfaces
I am working with closed degenerating hyperbolic Riemann surfaces, and I try to understand the compactification of the moduli space. Looking in different books, notably the one of Hummel, I now get a ...
- 914
9
votes
1
answer
3k
views
Oloid and sphericon: rolling develops entire surface
Wikipedia says that,
"The oloid is one of the only known objects, along with some members of the sphericon family, that while rolling, develops its entire surface."
Below are illustrations of ...
- 151k
3
votes
2
answers
348
views
Request for some references exploring the connections of Riemann surfaces with medical imaging
I'd like to know some references for a beginner who has basic background in Riemann surfaces and differential geometry, and would like to start learning/working on more applied areas, medical imaging/...
- 1,467
-1
votes
2
answers
1k
views
Regarding understanding differential geometry [closed]
I am essentially looking for a book that would hold my hand through basic concepts to more complicated ones. I am coming from physics. I am looking to make some connections with Classical mechanics ...
user39066
12
votes
3
answers
1k
views
A version of Lusternik–Schnirelmann category for good open covers
Recall that the Lusternik–Schnirelmann category (or LS-category) of a space is the integer $n$ such that there is an open cover by $n+1$ open sets which have nullhomotopic inclusions, and no such ...
- 35.5k
9
votes
2
answers
399
views
Reference for a path groupoid being a diffeological groupoid
I am looking for a reference that has a proof that a path groupoid
is a groupoid internal to the category of diffeological spaces. I do know how to prove this fact, and a proof is not hard. My reason ...
- 1,710
15
votes
2
answers
654
views
Invariant differential operators on real Grassmannians
I am looking for an explicit description of the algebra of $SO(n)$- or, better, $O(n)$-invariant differential operators on the real Grassmann manifolds of $k$-dimensional linear subspaces in the ...
- 21.8k
1
vote
1
answer
117
views
Minimal Legendrian submanifolds and laplacian of particular functions
I'm reading the paper
Lê, Hôngvân(D-MPI-NS); Wang, Guofang(PRC-ASBJ-MSY)
A characterization of minimal Legendrian submanifolds in $S^{2n+1}$. Compositio Math. 129 (2001), no. 1, 87–93.
Let $x: L^n \...
- 585
15
votes
0
answers
382
views
Has Cheeger's 'de Rham cohomology' of metric measure spaces been studied beyond its definition?
In J. Cheeger's 'Differentiability of Lipschitz Functions on Metric Measure Spaces' (Geometric and Functional Analysis, 1999, Vol. 9 pp 428-517, see here), a 'de Rham cohomology group' $H_{dR}^1(Z,\mu)...
user14166
11
votes
2
answers
1k
views
Diffeology as a sheaf on the site of smooth manifolds
Souriau's definition of diffeology may be phrased as defining a concrete sheaf on the category $\mathsf{Open}$ of open subsets of Euclidean/coordinate spaces. It seems to me, unless I am missing ...
- 1,710