# 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].

323
questions

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 := ...

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 ...

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 (...

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 ...

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 $\...

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 ...

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 \...

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 ...

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-...

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 ...

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 ...

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-...

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_{\...

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 ...

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 ...

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?

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 ...

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 $H$-...

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 ($x^a$ ...

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 ...

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. ...

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 ...

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 ...

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 ...

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 \...

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 $\mathscr{B}$...

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 ...

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 ...

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}\...

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 ...

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.
...

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. ...

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 ...

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 ...

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 ...

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 (...

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)...

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 ...

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 ...

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 $\...

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 ...

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 ...

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-...

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\...

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 ...

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 ...

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 (...

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 ...

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: ...