All Questions
Tagged with reference-request dg.differential-geometry
800 questions
6
votes
2
answers
317
views
Quasi-isometric embedding of graphs in non-compact riemannian surfaces
Given a complete riemannian surface $(S,m)$, where $S$ is homeomorphic to $\mathbb{R}^2$, I would like to find a weighted graph $G$ (which means a graph with real non-negative weights on the edges), ...
- 935
6
votes
1
answer
870
views
Physical (GR) Differential Geometry?
I am looking for problem lists or books which contain open problems in the area of mathematics motivated by physics. Ideally, I am looking for questions asking about which reduce to some calculation ...
- 191
6
votes
1
answer
342
views
The Cauchy problem in general relativity, hyperbolic PDEs, and Sobolev spaces on manifolds
(I apologize in advance if this question is ill-posed or not suitable for Math Overflow, I am not yet a research mathematician, just a student.)
Let $(\Sigma,\bar{g})$ be an $n$-dimensional Riemannian ...
- 221
6
votes
1
answer
1k
views
Geometry of the complex quadric
The complex orthogonal group $O(n+1, \mathbb{C})$ acts transitively on the complex quadric
$$
Q_{n-1} := \{[z_0:z_1: \cdots :z_n] : z_0^2 + \cdots z_n^2 = 0 \} \subset \mathbb{CP}^n.
$$
What is ...
- 13.5k
6
votes
2
answers
1k
views
Shuffle (co-)multiplication and generalized Leibniz formula in tensor calculus
The headline already says it: Is anybody (except me, UPDATE: plus Gavrilov) aware of this formula for higher total covariant derivatives of tensor products?
It is the simplest application of the ...
- 1,068
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
6
votes
1
answer
522
views
Preprint of Hamilton on deformations of foliations
Does anyone have access to Hamilton's 1978 Cornell preprint 'Deformation Theory of Foliations'. It is widely quoted but I couldn't find any online copy.
- 183
6
votes
1
answer
185
views
Zeta-Determinant for shifted Laplacians on the circle
Consider on the circle $S^1$ the operator
$$L := - \frac{\partial^2}{\partial \theta^2} + c$$
for some constant $c \in \mathbb{R}$.
What is its $\zeta$-regularized determinant?
This should be well-...
- 9,853
6
votes
1
answer
1k
views
Basic results in bounded geometry
I'm doing analysis (dynamical systems) in the context of Riemannian manifolds of bounded geometry and I find myself reproving quite a few standard results/tools from standard differential geometry, ...
- 2,481
6
votes
1
answer
399
views
A possible generalization of the exponential map
Let $M$ be a $n$-dimensional Riemannian Manifold, fix $p\in M$, and $1<k<n$. Do we know if the following is true?
For any $k$-dimensional subspace $V$ of $T_p M$, there exists a minimal ...
- 1,630
6
votes
1
answer
417
views
"structure group" for fibration
Regarding "fibration" as a homotopy analogue of "fiber bundle",I want to see parallel notions of "structure group" and "fiber change" in "fibration".
Does it make sense to talk about "structure group"...
- 71
6
votes
0
answers
132
views
Generalization of pseudogroups
Pseudogroups are defined here: https://ncatlab.org/nlab/show/pseudogroup
One of the problems with defining manifolds in terms of pseudogroups is that it gives no notion of a morphism between manifolds,...
- 181
6
votes
0
answers
122
views
Given the Ricci decays rapidly to 0 at infinity, is the metric asymptotically flat?
Consider the manifold $M=\mathbb{R}^3 \setminus B$ where B is the ball with radius 1. Let $f \in C^{ \infty}(M) $ satisfying:
$$f = \frac{C(\theta, \phi)}{r} + O( r^{-2}) $$
Where $(r,\theta,\phi)$ ...
- 969
6
votes
0
answers
355
views
Higher order variations of Riemannian geodesics
Consider a mapping $\Gamma$ from the Euclidean plane or an open subset to a Riemannian manifold $M$ so that each $\Gamma(s,\cdot)$ is a geodesic.
There is a well established theory of the first order ...
- 8,086
6
votes
0
answers
163
views
Reference request: normal form of k-differentials and flat surfaces at a puncture
Let $f(z)$ be a holomorphic function defined on a punctured neighborhood of $z=0$ with non-essential singularity of degree $d$ at $0$ (namely, $f(z)=z^dh(z)$, where $h(z)$ is a holomorphic function ...
- 1,804
6
votes
0
answers
388
views
What’s the limit of a vector bundle?
In geometric measure theory, there’s an answer to the question “what’s the limit of a family of submanifolds”, namely there’s some kind of object called an integral current.
In the geometric ...
- 8,723
6
votes
0
answers
270
views
Varying a $J$-holomorphic sphere in a symplectic $4$-manifolds
I am certain that the following result holds, but was not able to find a reference. Do you know one? Or maybe you can give a short proof?
Statement. Let $(M^4,\omega)$ be a compact symlectic manifold ...
- 14.3k
6
votes
0
answers
239
views
Complex submanifolds via Kähler reduction
Let $X$ be a Kähler manifold with an isometric $S^1$-action (which of course complexifies to a $\mathbb C^*$ action). Consider the corresponding Hamiltonian $H$ and let $X_0=H^{-1}(0)/S^1$ be the ...
- 14.3k
6
votes
0
answers
493
views
Reference for the Banach Manifold structure of $C^k(M,N)$
I'm completely new to the subject of banach manifolds and I'm looking for a reference of the following:
Let $M$,$N$ be smooth (=$C^\infty$) finite-dimensional compact manifolds. Consider the set $C^...
- 71
6
votes
0
answers
352
views
How to generate a random (Weyl) curvature operator ?
Given a dimension $n$, the space of curvature operators is the space $S^2_B(\Lambda^2\mathbb{R}^n)$ of symmetric endomorphisms $R$ of $\Lambda^2\mathbb{R}^n$ which satisfy the first Bianchi identity :
...
- 4,123
6
votes
0
answers
437
views
Has anyone seen this Hitchin-like system?
Let $(M,g)$ be a riemannian manifold and let $P\to M$ be a principal $G$-bundle with connection $A$. Let $\alpha \in \Omega^1(M;\mathrm{ad}P)$ be a one-form on $M$ with values in the adjoint bundle $\...
- 33.3k
5
votes
2
answers
2k
views
Canonical reference for Chern characteristic classes
I'm a little uncertain about the definitions for
Chern roots
Chern classes
Chern characters
From perusing several discussions, I gather that if one correlates the nomenclature with that of ...
- 10.5k
5
votes
4
answers
954
views
literature on geometrical viewpoint on calculus of variations for physics
What is a good reference for a geometrical viewpoint on the calculus of variations for physics, using differential forms etc. to derive Yang-Mills equations and other topics of the standard model?
...
- 921
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
5
votes
1
answer
8k
views
Is every real vector bundle over the circle necessarily trivial?
Is every real vector bundle over the circle necessarily trivial? If yes - could you please point to a reference. If no - what are sufficient conditions?
I am particularly concerned with the case of a ...
- 2,935
5
votes
2
answers
307
views
Compact surface with arbitrarily large eigenvalue
Consider a compact surface $M$ with genus $\gamma \geq 2$ and fix a positive real number $V$. Is it known whether it is possible to produce a metric $g$ on the surface $M$ such that $(M. g)$ has ...
- 51
5
votes
3
answers
1k
views
Constant rank theorem for Banach spaces
Is there a similar statement to the constant rank theorem for finite dimensional real smooth manifolds which holds for a smooth map $F:B \rightarrow M$ where $B$ is an infinite (countable) dimensional ...
- 2,099
5
votes
3
answers
892
views
First Minkowski Formula
Does anyone know of a modern proof of the First Minkowski Formula for a compact embedded hypersurface $\psi \colon \mathcal{M}^n \hookrightarrow \mathbb{R}^{n+1}$ ? The integral formula is
$$ \int_{\...
- 1,123
5
votes
1
answer
1k
views
An almost complex structure on the real $n$-sphere $S^n$
If $R\mathrel{:=}\mathbb{R}[x_1,\dotsc,x_{n+1}]/(x_1^2+\dotsb+x_{n+1}^2-1)$ and $S^n\mathrel{:=}\operatorname{Spec}(R)$ is the real $n$-spere, a classical result of Borel and Serre says that the only ...
user122276
5
votes
1
answer
202
views
Forms satisfying the zero-energy condition on the projective plane
Theorem (Michel). A $1$-form on the projective plane is exact if and only if its integral over any projective line is equal to zero.
Is there a simple proof of this result due, I think, to R. Michel ?...
- 13.5k
5
votes
3
answers
550
views
Can the conformal structure on the projective light-cone detect hyperplane sections?
Let $(V,\langle\,\cdot\,,\,\cdot\,\rangle)$ be an $(n+1)$-dimensional real vector space, equipped with a nondegenerate symmetric bilinear form of indefinite signature, and denote by $\nu(v):=\langle v,...
- 2,527
5
votes
1
answer
954
views
Literature Request: Berger Spheres and their Construction
In a previous question by me I asked about Berger spheres and their Lorentzian analogue, squashed $AdS_3$ along Hopf fibres. It was well answered (by https://mathoverflow.net/users/13268/ben-mckay) ...
- 679
5
votes
1
answer
577
views
Isomorphic algebras determine diffeomorphic manifolds
It is a kind of folklore but I would like to see the proof of the following fact: given two smooth manifolds $M$ and $N$ if we assume that the algebras $C^{\infty}_0(M)$ and $C^{\infty}_0(N)$ are ...
- 243
5
votes
1
answer
482
views
Besse p134 Riemann tensor in dimension 4
Does someone have a reference for the proof of 4.72 page 134 of Einstein Manifolds? It is said that
$$\check{R}-\vert R\vert^2g/4=S/3 (Ric-S/4) +2\mathring{W}(Ric -S/4) $$
because we are in dimension ...
- 914
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
5
votes
1
answer
343
views
Clarifying a result of Klingenberg
I am looking for some clarification on a result of Klingenberg. For context, I have a complete Riemannian manifold for which I would like to compute a lower bound on the injectivity radius at each ...
- 163
5
votes
1
answer
306
views
Topological invariance of rational Pontrjagin classes for non-compact spaces
Given a homeomorphism between complex manifolds, $f : X → Y$, is it then true that the rational Pontrjagin class $p_1(X) \in H^4(X,\mathbb Q)$ equals the pull-back $f^* p_1(Y)$?
If $X$ and $Y$ are ...
- 153
5
votes
1
answer
594
views
Existence of nonvanishing Killing field
Let $(M,g)$ be a closed Riemannian manifold.
Q Is there any research about the existence of nonvanishing Killing field, especially the nontrivial example.
- 1,915
5
votes
2
answers
569
views
3-sphere bundles over 4-sphere bound smooth disc bundles
I saw in the answer of this post:
Is it true that all sphere bundles are boundaries of disk bundles?
that a $S^3$-bundle over $S^4$ bounds a disc bundle over $S^4$ iff $O(4)\rightarrow Diff(S^3)$ is ...
- 163
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
5
votes
4
answers
2k
views
Relationship between the focal locus and the cut locus
I am seeking
clarification of
the relationship between the
focal locus
and the
cut locus
of a curve $C$ in $\mathbb{R}^2$, and
of a surface $S$ in $\mathbb{R}^3$.
Essentially my question is,
Under ...
- 151k
5
votes
1
answer
407
views
Making a submanifold transverse to a vector field by an isotopy
Let $M$ be a smooth manifold, $N\subset M$ be a smooth closed hypersurface not bounding a compact submanifold, and $X$ be a smooth nowhere-zero vector field on $M$. I would like to learn what is known ...
- 501
5
votes
1
answer
153
views
An estimate on deviation of two smooth tangent $J$-holomorphic curves
Take $\mathbb C^2$ with coordinates $(z,w)$. Suppose that $J$ is a $C^{\infty}$ almost complex structure on $\mathbb C^2$ such that the line $w=0$ is $J$-holomorphic and $J(0,0)$ is given by $(z,w)\to ...
- 14.3k
5
votes
1
answer
1k
views
On the complexification of a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle. If we suppose $TM\otimes\mathbb{C}$ is the complexification of $TM$ then how can we define a natural metric on the complex bundle $...
- 601
5
votes
1
answer
267
views
Generalized Plateau problem with non-Jordan boundary
Let $C_\pm$ be the two circles obtained by intersecting the cylinder $x^2+y^2=R^2$ with the planes $z=\pm 1$, on which we mark four points $A_\pm:(R,0,\pm 1)$ and $B_\pm:(-R,0,\pm 1)$. Assume that $R$...
- 2,581
5
votes
1
answer
234
views
Does $F_{A}^{0,2}=0$ for a connection $A$ on $TM$ almost complex give a complex structure?
First, some motivation. Let $X$ be a complex manifold, and $A$ a Hermitian connection on some complex vector bundle $E$ over $X$. It is known that the existence of $A$ such that the $(0,2)$-part of ...
- 1,763
5
votes
1
answer
224
views
Spectral theory of infinite volume hyperbolic manifolds
I have a question about the discrete spectrum of the Laplace operator on hyperbolic manifolds with infinite volume. I understand the case of infinite area surfaces: see Chapter 7 (Sections 1 and 2) of ...
- 1,407
5
votes
1
answer
743
views
Eigenvalues and Domain of the Laplace-Beltrami Operator
Assume $(M,g)$ is a compact Riemannian manifold without boundary, where $g$ is the Riemannian metric. Let $L:=-\Delta$ be the Laplace-Beltrami operator on $M$ defined by $\Delta \cdot = \text{div}(\...
- 51
5
votes
1
answer
413
views
Casson invariant and Euler characteristic
A slogan I frequently hear is: "the Casson invariant is the Euler characteristic of the Floer homology of flat SU(2)-connections on the integral homology sphere". Is there a single paper/reference ...
- 31
5
votes
1
answer
978
views
Existence proof of Bourbaki, Differentiable and Analytic Manifolds
I am reading through Chapter III of Bourbaki, Lie Groups and Lie Algebras, and many proofs cite the Bourbaki volume Differentiable and Analytic Manifolds. I can't find this book anywhere. Does it ...
- 6,180