All Questions
Tagged with reference-request dg.differential-geometry
800 questions
7
votes
1
answer
296
views
ASD connection for Line bundle over $4$-manifold
Let $(M,g)$ be an oriented closed Riemannian $4$ manifold.
Let $L\to M$ be a complex line bundle.
Q Under what condition, we can find an ASD connection of $L$, i.e. a connection $A$ such that $F^+...
- 1,915
7
votes
1
answer
612
views
Foliation with trivial leaf holonomy
In 1960, R. Hermann showed the following:
Theorem Let $M$ be a manifold with a foliation $F$ and a bundle-like metric, if all leaves are compact and the holonomy group of each leaf is trivial, then $...
- 1,915
7
votes
1
answer
554
views
Lower bound on $L^2$ norm of mean curvature in general dimensions
Suppose $\Sigma\subset \mathbb{R}^{n+1}$ is a closed embedded hypersurface. We know that when $n=1$
$$
\int_{\Sigma} |H|^2 \geq \frac{4 \pi^2}{|\Sigma|}
$$
by Gauss-Bonnet and that this is saturated ...
- 2,299
7
votes
1
answer
232
views
Connection of principal fiber bundles — history
I wonder who was the first to discover the notion of principal fiber bundle and its connection (gauge field in the physical language). Wikipedia cites the book by Steenrod (1951). But was he the ...
- 183
7
votes
1
answer
210
views
Recognizing sections up to isotopy
Let $E$, $B$ be smooth manifolds, $\pi\colon E\to B$ be a smooth fiber bundle, and $h:B\to E$ be a smooth embedding. I would like to learn what is known about the following
Question. When does there ...
- 501
7
votes
1
answer
723
views
Complex manifolds with corner?
I was reading Dominic Joycee article on Manifold with corner. He talk about manifold with corner modeled over $[0,\infty)^k\times \mathbb R^{n-k}$ for some $k\leq n$. From here I moved to Melrose ...
- 227
7
votes
1
answer
1k
views
Helmholtz-Decomposition on compact Riemannian manifolds
For smooth domains $\Omega$ in $\mathbb{R}^n$ it is known that one can decompose vector fields in $L^p(\Omega)^n$, $1 < p <\infty $ into a "gradient"- and a "divergence-free"-part such that
$L^...
- 73
7
votes
2
answers
359
views
Cone unfolding of space curves
There is a natural length-preserving operation which transforms any rectifiable space curve $\gamma\colon [a,b]\to R^n$ into a planar curve $\tilde\gamma \colon [a,b]\to R^2$. This operation, which ...
- 7,169
7
votes
1
answer
257
views
Bimeromorphic equivalence of reduced spaces for Kähler $S^1$-actions
Let $(X,\omega)$ be a smooth Kähler manifold (not necessarily compact) with an isometric $S^1$-action with a Hamiltonian $H$. It is a well known fact that
1) The reduced spaces $X(c)=H^{-1}(c)/S^1$ ...
- 14.3k
7
votes
1
answer
455
views
Properties of the affine curve-shortening flow
The affine curve-shortening flow (ACSF) seems to satisfy the following properties, even if the initial curve(s) is (are) self-intersecting (at least as long as they are "nice enough"):
(1) The length ...
- 1,174
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
496
views
Finite atlas on a smooth manifold
If a smooth manifold admits a finite atlas, then for many technical purposes it is as good as compact. I was surprised to learn that every connected smooth manifold actually has a finite atlas. This ...
- 1,959
7
votes
0
answers
236
views
Chern-Weil theory on some noncompact groups, and characteristic classes in differential cohomology
$\newcommand{\Z}{\mathbb Z}\newcommand{\HdR}{H_{\mathrm{dR}}} \newcommand{\Sym}{\mathrm{Sym}}
\newcommand{\g}{\mathfrak g}$I have a specific question about invariant polynomials for some Lie groups,
...
- 6,881
7
votes
0
answers
248
views
Does the Hodge decomposition hold for equivariant differential forms?
Let $M$ be a Riemannian manifold. The Hodge decomposition tells that
$$
\Omega^*(M) = \mathrm{im} \ d \oplus \mathrm{im} \ d^* \oplus \mathscr H^*(M)
$$
where $d^*$ is the adjoint operator of the ...
- 2,789
7
votes
0
answers
265
views
Generalized differential geometry based on Penrose's abstract tensor systems?
Penrose graphical notation has been an important precursor of string diagrams for monoidal categories. It was introduced in Penrose's paper Applications of negative dimensional tensors with intended ...
- 6,406
7
votes
0
answers
407
views
Linear vs smooth actions of finite groups on spheres, euclidean spaces and closed disks
I would like to know examples (with references, if possible) of the following:
(1) a finite group $G$ acting effectively and smoothly on a sphere $S^n$ (any $n$) but admitting no effective linear ...
7
votes
0
answers
116
views
Bundles over Function Spaces
Is there any reference on bundles over function spaces? In particular, I am interested in Banach-bundles over function spaces like $W^{k,r}(M)$, where $M$ is a Riemannian manifold. Separable Hilbert-...
- 71
7
votes
0
answers
1k
views
Closed geodesics on a closed, negatively curved Riemannian manifold
I have been searching for a while for a proof of the following fact: For a closed Riemannian manifold, all of whose sectional curvatures are negative, each free homotopy class of loops contains a ...
- 71
6
votes
2
answers
755
views
Topology/geometry of $O(2n)/U(n)$
I am interested in what is known about the topology (diffeomorphism type) or geometry (is it complex? non-complex? symplectic?) of either the compact space of orthonormal complex structures on $\...
- 323
6
votes
3
answers
1k
views
Where to find the results of Onishchik?
I would like to have a good reference where the results in
"Inclusion relations between transitive compact transformation groups" https://mathscinet.ams.org/mathscinet-getitem?mr=27:3740
can be ...
- 73
6
votes
2
answers
2k
views
How many metrics of constant curvature exist on a Riemannian surface?
I have been trying to determine the number of metrics of constant curvature on a surface of genus $n$, say $\Sigma$. For low values, the answer is clear, the moduli space is a point for the sphere, ...
- 841
6
votes
2
answers
701
views
Lower regularity version of Moser's theorem on volume elements
A theorem of Moser, published in "On the Volume Elements of a Manifold" (Transactions of the Americal Mathematical Society 120, 1965; doi: 10.1090/S0002-9947-1965-0182927-5, jstor), shows that if a $C^...
- 23.2k
6
votes
5
answers
3k
views
Navier-Stokes equations in Riemannian geometry
The Navier-Stokes equations can be written on a Riemannian manifold as:
$$\dot{u}+\nabla_u u+ \Delta u=(df)^* $$
$$d^* u=0$$
where $\nabla$ is the Levi-Civita connection, $u$ is a vector field, $\...
- 663
6
votes
2
answers
1k
views
Parallel forms and cohomology of symmetric spaces
Let $G/H$ be a compact symmetric space. Then I believe the following is true: if $\alpha \in \Omega^k(G/H)$ and $\nabla$ the Levi-Civita connection, then
$$
(\alpha \text{ is induced by an $\...
- 3,244
6
votes
2
answers
381
views
Sources for Alexandrov surfaces
There are two distinct notions in differential geometry associated
with A. D. Alexandrov: (1) Alexandrov spaces of courvature bounded
from below; (2) Alexandrov surfaces of bounded total curvature (...
- 16.6k
6
votes
2
answers
706
views
Reference request: uniformization theorem
I would appreciate if someone could point me to some introductory literature/resources where I can learn about Poincaré's uniformization theorem at a basic level.
Any good powerpoint notes, short ...
- 63
6
votes
3
answers
590
views
When does one obtain different 3-manifolds by pasting two tori?
Consider a compact solid torus $T$ and a diffeomorphic copy of it $T' \subset T$ embedded in the interior of $T$ in such a way that it makes two turns around the central circle of $T$.
I would like ...
- 4,304
6
votes
2
answers
926
views
Learning roadmap to 'Differential cohomology in a cohesive $\infty$ topos'
I am very curious to study arXiv:1310.7930 (henceforth:DCCT) but am not sure if I have the pre-requisites. I am familiar with basic algebraic topology (singular cohomology, classifying spaces, ...
- 709
6
votes
1
answer
229
views
Does $\pi_k(M)\neq 0$ implies $\operatorname{ind}(\gamma) < k$?
Cross post from MSE. and sorry if this is an obvious question.
Here is a line of proof of Theorem 1.15 from
Brendle, Simon, Ricci flow and the sphere theorem, Graduate Studies in Mathematics 111. ...
- 4,195
6
votes
2
answers
339
views
Continuity of the spectrum with respect to the metric
The following question is quite natural, but I am not aware of a reference dealing with it: let $M$ be a compact (smooth) manifold (posssibly with boundary) and $E$ a vector bundle on $M$ with an ...
- 3,399
6
votes
3
answers
561
views
Smale's theorem for $C^1$ diffeomorphisms of the sphere
In 1926 Kneser showed that homeomorphisms of $\mathbf{S}^2$ admit a retraction into the orthogonal group $O(3)$. Smale extended this result to Diffeomorphisms of $\mathbf{S}^2$ in 1958; however, in ...
- 7,169
6
votes
2
answers
2k
views
Line bundles over Kähler–Hodge manifolds
A Kähler–Hodge manifold $M$ can be defined as a Kähler manifold whose Kähler form $\omega$ is integral, namely $\omega\in H^{2}(M,\mathbb{Z})$. It is known then that there always exists a Hermitian ...
- 2,816
6
votes
1
answer
160
views
Metrics of non-negative sectional curvature on $S^7$-bundles over $S^8$
In Curvature and symmetry of Milnor spheres, Grove and Ziller construct metrics of non-negative sectional curvature on $S^3$-bundles over $S^4$ (by using a cohomogeneity one action). In the same paper,...
- 641
6
votes
1
answer
468
views
Characterization of bounded geometry - Reference-request
I already asked this question at stackexchange three days ago. Since I got no answer, I want to try mathoverflow now. I hope that you can help.
I'm looking for a proof of an equivalence that can e.g. ...
- 63
6
votes
3
answers
889
views
Reference request: embedded Morse theory
For references of embedded Morse theory, or so-called relative morse theory of a pair, I have found R.W. Sharpe's "Total Absolute Curvature and Embedded Morse numbers" totally not helpful. For further ...
- 2,274
6
votes
1
answer
342
views
Contracting a geodesic on a space of curvature less than 1
I would like to ask for a reference to the following statement (hopefully correct):
Let $M$ be a manifold of sectional curvature at most $1$ and let $\gamma$ be a closed geodesic.
Suppose that $\...
- 28.9k
6
votes
3
answers
1k
views
The isometry group of a product of two Riemannian manifolds
Under what conditions is the isometry group of a product of two Riemannian manifolds the product of the isometry groups of each one of the components?
One counterexample is a product of two isometric ...
- 2,535
6
votes
2
answers
457
views
Symplectic orthogonality and projective duality: how do they work together?
Let $(V,\omega)$ be a $2n$-dimensional linear symplectic space, and $(\mathbb{P}V,\theta_\omega)$ the corresponding $(2n-1)$-dimensional contact manifold.
Given a smooth $(n-1)$-dimensional smooth ...
- 2,527
6
votes
2
answers
461
views
How many minimal surfaces do we have if the metric in the target space is not flat
It is trivial that there are a lot of minimal surfaces in the flat $R^3$: for example, for any point, any 2-plane containing this point,
and any two othogonal vectors in this plane, and any ...
- 4,787
6
votes
2
answers
3k
views
References for the Poincaré-Cartan forms
Hello, everybody. I'm looking for some reference about the Poincaré-Cartan form, I do not know how it is defined, I just know that it is used in Lagrangian mechanics but I have not found any ...
- 463
6
votes
1
answer
816
views
Proof that every three-dimensional Einstein manifold has constant curvature
In pseudo-Riemannian geometry it is well known that every three-dimensional Einstein manifold has constant curvature. A proof of this is sketched here.
Question. Does anyone know where in the ...
- 985
6
votes
2
answers
448
views
About the index theorems
I am looking for some introductory book/paper/notes about the several index theorems and their applications. By several I mean the "classical" Atiyah-Singer theorem, the local index theorem (...
- 789
6
votes
1
answer
1k
views
Fubini's theorem on arbitrary foliations
In what follows $ \mathbb{R}^{n+m} = \{(x,y): x \in \mathbb{R}^n, \ y \in \mathbb{R}^m \} \ .$
Suppose $G: U \to V $ is a $C^1$-diffeomorphism from an open subset of a manifold to an open subset of $...
- 622
6
votes
1
answer
342
views
Can the number of solutions to a system of PDEs be bounded using the characteristic variety?
I've recently come across a system of PDEs which I'd like to understand better. The particular system I'm interested in locally solves for a 2-dimensional Riemannian metric as the Hessian of a ...
- 6,001
6
votes
1
answer
260
views
Fundamental class in $KO[1/2]$
Let $M^m$ be an oriented Riemannian manifold.
The signature operator associates to $M$ a class $\Delta_M\in KO_m(M)[1/2]$.
I have two questions about this class $\Delta_M$:
Rationally, $\Delta_M$ is ...
- 61
6
votes
2
answers
446
views
Is every $S^3$ block bundle over $S^4$ a fiber bundle?
I am interested in the difference between block bundle and fiber bundle.
Let $K$ be a simplicial complex and $p: E\to |K|$ be a continuous map.
A block diffeomorphism of $\Delta^p\times M$ is a ...
- 73
6
votes
1
answer
723
views
Geometric treatment of the Ward-Takahashi identity
The quantum field theory generalisation of Noether's theorem about symmetries and conservation laws is the Ward-Takahashi identity.
What is a suitable treatment of this in the context of differential ...
- 921
6
votes
1
answer
207
views
Coarse embeddings and Gromov products in (Gromov) hyperbolic spaces
I am new into geometric group theory and I have recently started reading the book "Sur les Groupes Hyperboliques d’après Mikhael Gromov" by Ghys and de la Harpe. The following inequality ...
- 101
6
votes
1
answer
423
views
Difference between parallel transport and ambient projection
Consider a $d$-dimensional complete embedded Riemannian submanifold $(M,g)$ of a Euclidean space $\mathbb{R}^D$ (The major examples we consider are sphere and Stiefel manifold). Assume the sectional ...
- 125
6
votes
1
answer
439
views
Name for a class of almost symplectic manifolds
A $2n$-dimensional manifold $M$ is said to be almost symplectic if it possesses a non-degenerate two-form $\omega \in \Omega^2(M)$. Equivalently, an almost symplectic structure is a $G$-subbundle $P \...
- 33.3k