Skip to main content

Questions tagged [connections]

Ehresmann connections; covariant derivatives; connections on vector bundles, principal bundles, ∞-bundles, submersions, bundle gerbes; holonomy and higher holonomy; parallel transport; torsion; curvature. See also the tags [principal-bundles], [vector-bundles], [gerbes], [curvature], [geodesics], [characteristic-classes], [torsion].

Filter by
Sorted by
Tagged with
4 votes
0 answers
86 views

Can we get a connection on the principal bundle from a connection on the associated vector bundle?

Assume $G$ is a Lie group, $P \to M$ is a smooth principal $G$-bundle, and $\rho \colon G \to GL(V)$ is a smooth representation of $G$. We can define a connection on the associated vector bundle $E := ...
mfdmfd's user avatar
  • 99
3 votes
0 answers
80 views

Torsion in terms of parallel transport

This MO answer establishes the connection between parallel transport and torsion in the special case of the canonical flat connections on a Lie group. This suggests a more general construction for an ...
Alex Bogatskiy's user avatar
2 votes
0 answers
47 views

Connection vs Exponential preserving maps

Connection Preserving Diffeomorphisms The setting is a manifold $M$ equipped with a linear connection $\nabla$. Kobayashi & Nomizu [K&N §VI.1] define a connection preserving diffeomorphism (...
Olivier's user avatar
  • 243
0 votes
1 answer
214 views

Atiyah sequence of a coherent sheaf

I want to show that if $X$ is a smooth complex projective variety, then analytification induces an equivalence of categories between algebraic integrable connections on $X$ and analytic integrable ...
Tanny Sieben's user avatar
5 votes
1 answer
170 views

Converging paths implies converging parallel transports along those paths?

Suppose we have a vector bundle $E$ with connection $\nabla$ over a smooth manifold $M$. Let’s also say we have a sequence of smooth paths $\gamma_n\in C^\infty([0,1],M)$ starting at the same point $\...
user815293's user avatar
1 vote
0 answers
45 views

Frobenius pullback of an integrable connection on a quasi-projective scheme

Let $X_k$ be a smooth quasi-projective scheme over a finite field $k$. Let $X_K$ be a smooth lift of $X_k$ to characteristic $0$, and let $(X_K)^{\text{an}}$ denote the rigid analytic space associated ...
kindasorta's user avatar
  • 2,103
1 vote
0 answers
65 views

Extending $G$-equivariant local diffeomorphisms on principal bundles to local bundle maps

Consider a principal $G$-bundle $P$ over the base space $M$ equipped with a connection 1-form $\omega$. Let $\mathcal{U}$ and $\mathcal{V}$ be open subsets of $P$, and suppose $F: \mathcal{U} \to \...
Amin's user avatar
  • 83
1 vote
1 answer
109 views

Frobenius action on the trivial connection

Let $F$ denote the absolute Frobenius acting on a smooth quasiprojective scheme $X$ over a finite field $k$. Denote the trivial connection on $\mathcal{O}_X$ by $d$. Denote its pullback by Frobenius ...
kindasorta's user avatar
  • 2,103
1 vote
0 answers
59 views

$F$-structure implies regular singularities + unipotent local monodromy?

Let $(\mathcal{E},\nabla)$ be a vector bundle with an integrable connection on a smooth quasi-projective $K$ scheme $X$, with $K$ a $p$-adic number field of characteristic $0$. Let $F$ denote a semi-...
kindasorta's user avatar
  • 2,103
1 vote
0 answers
50 views

Frobenius acting by autoequivalence on $\text{Isoc}(X/K)$

Let $X_k$ be a smooth quasiprojective scheme over a finite field $k$. Let $X_K$ be a smooth lift of $X_k$ to the fraction field of the Witt ring of $k$, which I denote by $K$. In various papers I read ...
kindasorta's user avatar
  • 2,103
4 votes
1 answer
247 views

Equivalence between vector bundles with integrable connections to isocrystals

Let $k$ be a perfect field, $W(k)$ its Witt ring, and $K$ the fraction field of $W(k)$. Let $X_k$ be a smooth proper curve over $k$, and let $X_K$ be the schematic generic fibre of a smooth proper ...
kindasorta's user avatar
  • 2,103
1 vote
0 answers
77 views

Are coherent modules with integrable log-connections locally free?

Let $X$ be a smooth Noetherian scheme over a field $K$. It is known that every coherent module with integrable connection on $X$ is locally free. Is the same true for coherent modules with log-...
kindasorta's user avatar
  • 2,103
2 votes
0 answers
392 views

Parallel transport on a vector bundle : expansion of the correspondance between normal and tubular coordinates

Let $(M, g^{TM})$ be a Riemannian manifold of dimension $n$. Let $X \subset M$ be a submanifold of dimension $n$ with boundary $\partial X$. Then we have a splitting of the tangent bundle $$TM \vert_{\...
hseldon39's user avatar
4 votes
0 answers
99 views

category of vector bundles with connections and its K-theory

For the category of Hermitian vector bundles with unitary connections, an object is (of course) a Hermitian vector bundle with a Hermitian metric and a unitary connection $(E, g^E, \nabla^E)$. For ...
Ho Man-Ho's user avatar
  • 1,117
4 votes
0 answers
145 views

Parallel transport of global sections and Riemannian curvature

A, perhaps, naive question from an algebraist/combinatorialist teaching differential geometry. Originally asked on math.SE but didn't receive a single comment in 3 days. Consider a (real) smooth ...
Igor Makhlin's user avatar
  • 3,503
0 votes
1 answer
182 views

Torsion free Chern connections and Kähler manifolds

Let $(M,h)$ be an Hermitian manifold and let $\nabla$ be the associated Chern connection. Is it true that $(M,h)$ is Kähler if and only if $\nabla$ is torsion free?
Zoltan Fleishman's user avatar
4 votes
1 answer
224 views

Let $G \subset \mathrm{GL}(n)$ be a matrix Lie group. Does there exist an affine connection under which the matrix and manifold exponential coincide?

Let $G\subset \operatorname{GL}(n)$ be a matrix Lie group. I am curious about curves $\gamma(t) = g \exp(tv)$, where $g \in G$, $v \in \mathfrak{g}$, and $\exp(.)$ is the matrix exponential. If we ...
Spencer Kraisler's user avatar
8 votes
0 answers
312 views

Flat Maurer-Cartan connection iff flat Berry connection

I am studying two connections on induced representation spaces $\text{Ind}_{H}^{G} \Gamma$, where $H \subseteq G$ are groups, and $\Gamma$ is an irrep of $H$. The first is the canonical or $H$-...
Victor V Albert's user avatar
1 vote
0 answers
143 views

Torsion free connection $\implies$ Jet coordinates $=$ Taylor expansion coefficients?

Suppose we have some smooth n-dimensional manifold $M$ endowed with basis 1-forms $\theta^a$ with $a=1\cdots n$. Then $\theta^a$ are sections of the coframe bundle $F^* M$. In local coordinates ($x^a$ ...
R. Rankin's user avatar
  • 230
5 votes
1 answer
259 views

Commutative/ symmetric second covariant derivative

Consider a smooth manifold $M$ together with an affine connection (or covariant derivative) $\nabla$ on the tangent bundle $TM$. Is it possible to have an affine connection, possibly with non-zero ...
Khaled T.'s user avatar
5 votes
0 answers
289 views

Connections in non-commutative geometry

Let $K$ be a field, $A$ a unital associative $K$-algebra and $M$ a left $A$-module. A connection on $M$ is a $K$-linear map $\nabla:M\to \Omega^1A\otimes_AM$ which satisfies the Leibniz rule. ...
Qwert Otto's user avatar
1 vote
1 answer
80 views

Chern character of a super-connection (Heat kernels and Dirac operators)

Let $A$ be a super-connection on a super-bundle $E\to M$, then the differential form \begin{equation} \mathrm{ch}(A)=\mathrm{Str}(e^{-A^2}) \end{equation} is called the chern character of $A$ on page ...
Filippo's user avatar
  • 329
1 vote
0 answers
130 views

Lifting action of torus to torus bundle

Preamble: Let $X$ be a simply connected smooth manifold and $P \to X$ be a principal $T^\ell$ bundle on it. Let $\phi$ be a smooth action of $T^k$ on $X$. The paper "Lifting compact group actions ...
Nicolò Cavalleri's user avatar
1 vote
0 answers
152 views

What does $\nabla^i f$ mean?

I am reading the article Some Geometric Calculations on Wasserstein Spaces of John Lott and there is this covariant index in the covariant derivative: $\nabla^i$. And I don't quite understand it. In ...
Gomes93's user avatar
  • 169
1 vote
1 answer
177 views

An identity for the higher form Levi-Civita connection

Take $M$ a Riemannian manifold and $\Lambda^1$ its space of one forms. The LCC (Levi-Civita connection) $\nabla:\Lambda^1 \to \Lambda^1 \otimes \Lambda^1$ is well known to satisfy the identity $m \...
Didier de Montblazon's user avatar
3 votes
2 answers
548 views

Obstructions to the existence of a flat connection on a vector bundle

Given a smooth manifold $M$ and a smooth vector bundle $E \to M$ (with real or complex fibers), what are known obstructions to the existence of a flat connection on $E \to M$? If all known ...
7 votes
1 answer
171 views

Homogeneous metric connections on 3-dimensional Lie groups

Let $G$ be a 3-dimensional unimodular Lie group equipped with a left-invariant metric $q$. Call $P_{SO}$ its oriented orthonormal frame bundle. Considering the moduli space of connections $\mathscr{B}$...
Matteo Bruno's user avatar
2 votes
0 answers
152 views

Understanding the Seiberg-Witten equations in dimension $3$

I am trying to understand the dimensional reduction of Seiberg-Witten equations from dimension $4$ to $3$, more specifically my concern is about ellipticity of the new equations in dimension $3$ under ...
Partha's user avatar
  • 893
1 vote
0 answers
72 views

The curvature of the induced connection on the antidual bundle

Let $E\to M$ be a complex vector bundle over a (real, smooth) manifold and $\nabla$ a connection on $E\to M$ whose curvature is $R$. From Section 1.5 of "Differential Geometry of Complex Vector ...
Ho Man-Ho's user avatar
  • 1,117
2 votes
1 answer
398 views

Bianchi's identity in a principal bundle

Let us consider a principal bundle $P$, with a Lie-algebra-valued connection one-form $\omega\in\mathfrak{g}\otimes\Omega^1(P)$ and a Lie-algebra-valued curvature two-form $\Omega\in\mathfrak{g}\...
Nabla's user avatar
  • 41
0 votes
0 answers
94 views

Affine manifold and topology

I consider an affine manifold $(M,\nabla)$, i.e. $\nabla$ is flat and torsion-free, such that: $M$ is diffeomorphic to $\mathbb R\times\Sigma$ with $\Sigma$ a closed 3-manifold. There exists a ...
Vigneron Quentin's user avatar
3 votes
0 answers
102 views

Geometric interpretation for a connection whose corresponding distribution generates the whole Lie algebras of vector fields

Let we have a connection on a manifold $M$ so it is considered as a distribution on the tangent bundle $TM$ of $M$. The integrability of this distrbution is equivalent to flatness of the connection. ...
Ali Taghavi's user avatar
3 votes
0 answers
219 views

What is a twisted D-module?

Let $X/\mathbb{C}$ be an abelian variety, $Y$ be the dual abelian variety, and $P$ be the Poincaré bundle on $X\times Y$. On p.207, Correction to “Sheaves with connection on abelian varieties” (by M. ...
Doug Liu's user avatar
  • 545
1 vote
0 answers
54 views

Extending connections defined on fibers to a connection defined over a fibration

Let $\pi:E\to B$ be a holomorphic fibration, and let $\mathcal{F}$ be a sufficiently nice sheaf (coherent for example) of $\mathcal{O}_E$-modules on $E$ that is flat over $B$ i.e. $\mathcal{F}$ is ...
Aidan's user avatar
  • 498
5 votes
1 answer
209 views

Orientation bundle and its flat connection

Let $M$ be a smooth $n$-manifold (which is not assumed to be orientable), and write $o(TM)\to M$ for its orientation bundle. Equivalently, it is the top exterior bundle $\Lambda^n(TM)\to M$. In any ...
Ho Man-Ho's user avatar
  • 1,117
3 votes
1 answer
142 views

When is compactness of fiber components an open condition?

Consider a smooth map $f:M\rightarrow N$ between smooth manifolds. Ehresmann's theorem states that if $f$ is a proper submersion, then it is locally trivializable; in particular, this implies that ...
Nikhil Sahoo's user avatar
  • 1,205
0 votes
0 answers
82 views

Confusion on a term related to connection and holonomy

This question is simply some of my confusions about a specific term. Let $E\to X$ be a trivial complex vector bundle. When one says let $\nabla^E$ be a connection on $E\to X$ with trivial holonomy (...
Ho Man-Ho's user avatar
  • 1,117
12 votes
3 answers
625 views

Modern treatment of Dirac monopoles and related topics

I know that the topic is classical and even "folklore", but many treatments make use of local coordinates and such treatments are rather messy. Could somewhere maybe provide some reference(s)...
Malkoun's user avatar
  • 5,118
3 votes
1 answer
97 views

Do we have an equivariant version of integrability theorem of flat connections?

I am reading Donaldson and Kronheimer's book The Geometry of Four-Manifolds. In page 48, I found Theorem 2.2.1: Let $H$ be the hypercube $H=\{\mathbf{x}\in \mathbb{R}^d|~|x_i|<1\}$. If $E$ is a ...
Zhaoting Wei's user avatar
  • 8,767
2 votes
0 answers
48 views

Gauge-natural lifts of principal connections

Let $P=(P,\pi,M,G)$ be a principal fibre bundle and $\omega$ a principal connection on it. If $\lambda:G\times S\rightarrow S$ is a smooth left action of $G$ on a manifold $S$, the associated fibre ...
Bence Racskó's user avatar
6 votes
0 answers
185 views

What is a non-smooth connection?

Let $p : E \to B$ be a map of topological spaces, and $p^I : E^I \to B^I$ the induced map of path spaces. Let $Cocyl(p) = B^I \times_B E$ be the space of paths $\beta$ in $B$ equipped with a lift of $\...
Tim Campion's user avatar
  • 62.6k
4 votes
1 answer
754 views

Coincide between Chern-connection and Levi-Civita connection

I am a beginner in complex geometry and I am going to show Levi-Civita connection $\nabla$ and the Chern connection $D$ are the same on the holomorphic tangent bundle $T^{1,0}M$ on Kahler manifold. By ...
James Chiu's user avatar
8 votes
2 answers
350 views

Given a Lie $2$-group $G$ does every principal $G$ $2$-bundle admit a $2$-connection?

The statement is true for Lie groups and principal bundles, with every principal bundle admitting a connection and I see no reason for the analogue result not to hold in the Lie $2$-group case but I ...
Eugenio Landi's user avatar
6 votes
1 answer
420 views

Holonomy bounded in terms of area and the curvature

I suppose the following result follows from Ambrose-Singer theorem, but I cannot find a reference, and the arguments I found in the literature are usually weaker. The idea is that holonomy over a null-...
Misha Verbitsky's user avatar
4 votes
1 answer
216 views

Existence of non-trivial "line-symplectic" manifolds

One way to view a symplectic manifold $(M,\omega)$ is as a real line bundle $\pi_1: M\times \mathbb{R}\to M$ equipped with a flat connection $d: \Omega^{k}(M, M\times\mathbb{R})\to \Omega^{k+1}(M, M\...
J.V.Gaiter's user avatar
3 votes
1 answer
269 views

Moduli space of flat connection over homology 3-sphere

I'm trying to understand the space of flat connections of the trivial $\mathrm{SU}(2)$-bundle over a closed, oriented homology three-sphere (for the purpose of understanding the instanton Floer ...
Lamda8's user avatar
  • 181
3 votes
0 answers
257 views

Manifolds and Riemannian geometry with a bundle viewpoint

I was wondering if there are any books that builds the theory on manifolds and Riemannian geometry, but at the same time treats these subjects in the general case of bundles (similar to Jeffery Lee's ...
Master.AKA's user avatar
3 votes
0 answers
121 views

Relation between $\mathrm{Pic}^\natural_{X/S}$ and two notions of rigidification

Let $X/S$ be a relative curve (perhaps with more adjectives). I have come across a few instances of rigidifying and rigidificators, which I would like to understand better. In Liu-Lorenzini-Raynaud (...
Somatic Custard's user avatar
6 votes
1 answer
452 views

What exactly is the relationship between an Ehresmann connection and splitting of the jet sequence?

An Ehresmann connection on a vector bundle $\pi : E \to X$ is a splitting of the sequence, $$ 0 \to V \to TE \to \pi^* TX \to 0 $$ which respects the linear structure on $E$ (meaning the section is ...
Ben C's user avatar
  • 3,373
11 votes
3 answers
829 views

Tangent bundle of a tensor product bundle

This question was also asked here on math-stackexchange. Let $E\to M$ and $F\to M$ be vector bundles. The structure of their tangents $TE$ and $TF$ is well known. In particular, connectors map $K_E: ...
Raz Kupferman's user avatar

1
2 3 4 5
7