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].

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3
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1answer
129 views

Curvature of principal bundle

Let $(P,M,G)$ be a principal bundle with connection 1-form $\omega$. In all books I have seen so far, the curvature is defined by \begin{equation} F:=D_{\omega}\omega \in \Omega({P,\mathfrak{g}}) \end{...
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0answers
40 views

Local coordinates of one form on a principal bundle

I am reading "Natural and Gauge Natural Formalism for Classical Field Theory" by Lorenzo Fatibene and I am really confused by his definition of a connection in local coordinates. Let's say ...
5
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1answer
442 views

Question about a proof in Berthelot's crystalline book

Below is an excerpt from Berthelot's book on crystalline cohomology. I don't understand the last sentence, namely why it follows that $\sigma\circ \varepsilon$ is an isomorphism. For what it's worth, $...
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0answers
71 views

Change in Connection on a complex Line bundle

Let's say $M$ is a compact Kähler manifold and $L$ is a complex line bundle on $M$. Now let's say $A$ be a connection or equivalently a hermitian metric on $L$. Hence one can have the operators $\bar\...
7
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0answers
158 views

(Higher) flat connections and Grothendieck construction

For any (nice) topological space there is an equivalence between covering spaces and local systems $\pi_1X \to \operatorname{Set}$. We can think of it as a Grothendieck construction. If we take ...
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0answers
61 views

About irreducible connection

The irreducible connection is a connection whose holonomy group is just $G$ (let us just assume the base space $X$ is just connected). Otherwise, it is called reducible if the holonomy group $H_A$ ...
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1answer
109 views

Projectively flat connection

Let $E \to B$ be a Hermitian vector bundle. If $E$ has a projectively flat connection, then its total Chern character has the form $\mbox{ch}(E) = \mbox{rank} \cdot \exp(\mbox{slope})$. Is the ...
2
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1answer
111 views

Splitting of higher order jet sequence

Let $X$ be a smooth variety. Because $\mathcal{O}_X$ admits a canonical connection $\mathrm{d} : \mathcal{O}_X \to \Omega_X$ the sequence, $$ 0 \to \Omega_X \to J^1(\mathcal{O}_X) \to \mathcal{O}_X \...
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2answers
578 views

When do flat holomorphic connections exist?

Let $X$ be a smooth projective variety over $\mathbb{C}$. I know that a vector bundle $\mathcal{E}$ on $X$ admits a holomorphic/algebraic connection iff its Atiyah class vanishes, $A(\mathcal{E}) = 0$....
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1answer
96 views

The notion of a “relatively” flat connection

Suppose that $X$ is a connected smooth manifold and $\Gamma$ is a group acting smoothly, freely, properly and discretely on $X$, so that $Y=X/\Gamma$ is another smooth manifold endowed with a covering ...
2
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1answer
106 views

Solving the Airy equation by Borel summation

The Airy equation is the canonical example of the Stokes phenomenon but, as of yet, I've not seen it being solved by Borel summation (which is the main way to explicitly construct examples of Stokes ...
4
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1answer
141 views

Curvature as infinitesimal holonomy 2

This question may be seen as a follow up of this original question. I'm learning Cheeger-Simons differential characters (reading Differential Characters of Bär and Becker). If I understand correctly, ...
1
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1answer
86 views

Flat connection of a degree zero line bundle on curve

The question is clear from the title. Suppose we have a line bundle on a compact smooth complex curve $X$, and a line bundle $\mathcal{L}=\mathcal{O}_X(p-q)$, where $p$ and $q$ are divisors, then what ...
4
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1answer
281 views

A de Rham space for meromorphic connections?

To any space $X$ you can associate its de Rham space $X_{dR}$. Vector bundles on $X_{dR}$ are the same thing as vector bundles on $X$ with a flat connection. Can anything like this be said for ...
17
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2answers
644 views

A non-Abelian de Rham complex?

This question is inspired by this physics stack exchange post, which is recent and has not received an answer yet, nontheless I feel that there is a better way to ask this question here with a larger ...
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0answers
96 views

Covariant Derivative of sections of a pullback bundle

Suppose that we have two smooth manifolds $M$ and $N$ and a smooth mapping $\phi : M \rightarrow N$. The Differential of that smooth mapping induces a bundle map $D\phi : TM \rightarrow TM$ between ...
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1answer
223 views

Yang-Mills over surfaces

For a principal bundle $(P,\nabla)\to M$ over a Riemann surface with fiber $G$, the Yang-Mills equation is $\nabla *F=0$, where $*F$ is dual of the curvature with respect to a fixed metric on the ...
2
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1answer
83 views

Almost geodesic on non complete manifolds

Let $M$ be a connected manifold equipped with a connection $\nabla$. By Hopf-Rinow theorem, we know that if $M$ is complete then for any $x,y$ there exist a curve $\gamma:[0,1] \to M$ such that $\...
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5answers
4k views

What is the Levi-Civita connection trying to describe?

I have seen similar questions, but none of the answers relate to my difficulty, which I will now proceed to convey. Let $(M,g)$ be a Riemannian manifolds. The Levi-Civita connection is the unique ...
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0answers
54 views

Curvature of a superconnection

Let $E\rightarrow X$ be a $\mathbb{Z}_2$-vector bundle (or superbundle for connoisseurs) and consider the superconnection $$A=\nabla + B$$ where $\nabla$ is a connection on $E$ and $B\in\Gamma(End(E))^...
3
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2answers
368 views

Pullback of a connection

let $X,Y$ be smooth schemes (or rigid spaces etc..) over a base $S$, let $f:Y \rightarrow X$ be a $S$-morphisn and let $\mathcal{F}$ be a locally free $\mathcal{O}_X$-module with connection $\nabla$. ...
6
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0answers
55 views

Is there a contact instanton connection on the tangent bundle of the 5-sphere?

A well-known example of a contact manifold is $S^5$, arising from it being a circle bundle over $\mathbb{CP^2}$. This is somewhat related to the reduction of the structure group from $SO(5)$ to $SU(2)$...
4
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4answers
541 views

Connections in the setting of algebraic geometry

My level is at the beginning of a second year master. I'm interested in the project of translating some features of differential geometry to algebraic geometry. I'd like to know if there is an ...
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1answer
186 views

What is the natural Lie groupoid structure on the Atiyah Lie groupoid of a principal $G$-bundle?

$\DeclareMathOperator\At{At}\DeclareMathOperator\Obj{Obj}\DeclareMathOperator\Mor{Mor}$According to https://ncatlab.org/nlab/show/Atiyah+Lie+groupoid#idea the Atiyah Lie groupoid $\At(P)$ of a ...
4
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2answers
354 views

Katz's proof of Cartier's (descent) theorem

I am trying to understand the proof of Cartier’s theorem on pages 370-371 (pages 17-18 of the PDF file) of Katz’s “Nilpotent connections and the monodromy theorem: applications of a result of ...
2
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1answer
96 views

Vector field along an immersion whose covariant derivative is the differential

Let $(M,g)$ be a Riemannnian manifold and let $f:\Sigma\to M$ be a smooth immersion. Then the vector bundle $f^\ast TM\to\Sigma$ has a natural bundle metric and metric-compatible connection. Can one ...
2
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2answers
198 views

Integrability condition for flat connections

I am reading Kobayashi's book "Differential geometry of complex vector bundles". More precisely, I am on section 2 of chapter 1, page 5. Kobayashi is trying to prove that if $E$ is a vector ...
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0answers
96 views

Can logarithmic connection on holomorphic vector bundle induce logarithmic connection on dual bundle?

Let $(X,\omega)$ be a compact K"ahler manifold of dimension $n$, and let $D=\sum_{i=1}^r D_i$ be a simple normal crossing divisor on it, i.e., a divisor with smooth components $D_i$ intersecting ...
3
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0answers
87 views

Opers and global differential operators

This is a follow up question to a previous question of mine and my thought of answer to it. Given a (compact) Riemann surface $\Sigma$, a $SL(n,\mathbb{C})$-oper is a rank $n$ holomorphic vector ...
6
votes
2answers
293 views

1d TQFT minus connection =?

Correct me if I am wrong but I believe at least conceptually (maybe even rigorously) data of a 1-dimensional TQFT and of a vector bundle with connection are equivalent. Going into more detail (and ...
4
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1answer
171 views

history of geometric mechanics

I was thinking about the foundations of geometric mechanics and its precursors. I wondered who was the first to realized the equivalence between Riemannian geometry and Lagrangian mechanics. In ...
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0answers
132 views

Examples of connection preserving maps in differential geometry

In synthetic differential geometry and tangent categories, linear connections on the tangent bundle are treated as a sort of algebraic gadgets that incorporate the tangent bundle. Like any other ...
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1answer
125 views

Simple example of non-integrable holomorphic connection

Let $X$ be a complex manifold with complex dimension $d$ and structure sheaf $\mathcal{O}_X$. Let $E$ be a locally free sheaf on $X$. A $holomorphic$ connection on $E$ is a morphism of sheaves $$\...
5
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0answers
190 views

Hopf fibration extended to bundle over $\mathbb{C}^2$

Consider the Hopf bundle $h:\mathbb{S}^3\rightarrow\mathbb{S}^2$. There is a connection $1$-form $\omega$ oh $h$ which is left $SU(2)$ invariant. In terms of the Euler angles $(\theta,\,\phi,\,\psi)$ ...
2
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1answer
213 views

Characterisation of (integrable) connections on (trivial) principal bundle

Let $M$ be a manifold. Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. Let $P(M,G)$ be a principal bundle. Recall that, a connection on $P(M,G)$ is a distribution $\mathcal{H}\subseteq ...
2
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1answer
189 views

Canonical connection on $\mathcal{A}\times X$

Let $E\rightarrow X$ be a vector bundle and let $\mathcal{A}$ denote the space of connections on $E$. Pulling back $E$ by the second projection we obtain a vector bundle $\mathbb{E}=p_2^*E\rightarrow ...
6
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1answer
238 views

What is the definition of homotopy flat connections?

What is a definition of a homotopy flat connection - in the context of differential forms with values in a dg algebra
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0answers
139 views

Geometric theory for cohomology groups $H^p(M;\mathbb{Z})$

An excerpt from the book Loop Spaces, Characteristic Classes and Geometric Quantization by Jean-Luc Brylinski is mentioned below: Characteristic classes are certain cohomology classes associated ...
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1answer
111 views

Connections on vector bundles over elliptic curves - concrete computations

This is linked to my question on math.Stackexchange for which I had no answer. I have found nowhere a historical account on connections (or rather called logarithmic connections?) that would help me ...
6
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0answers
149 views

Foliated circle bundles whose Euler class is torsion

Let $X$ be a closed manifold. By a foliated circle bundle $E \rightarrow X$ we mean a circle bundle over $X$ with total space $E$ and structure group $Diff^+(S^1)$, and a codimension one foliation of $...
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0answers
78 views

Does holonomy determine parallel transport? [duplicate]

Let $p: P \longrightarrow M$ be a smooth $G$-principal bundle endowed with a connection that determines the holonomies: $$\Phi_{\gamma}: P_{x} \overset{\cong}{\longrightarrow} P_{x}$$ for any fiber ...
2
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0answers
131 views

Pullback connection and diffeomorphism of the base

Let $p \colon E \to B$ be a vector bundle, $\nabla^E \colon E \to E \otimes \Omega^1_B$ a connection on $E$, and $\phi \colon B \xrightarrow{\sim} B$ a diffeomorphism. Further, let there be a natural (...
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1answer
266 views

Katz's paper on $p$-curvature – help with proof understanding

I am studying N. Katz's paper "Nilpotent connections and the monodromy theorem: applications of a result of Turrittin" where I found a fairly good account on $p$-curvatures. I don't understand the ...
2
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0answers
36 views

Solving equations of motion of holomorphic BF theory - pure gauge in complex coordinates

In this paper by Bailieu and Tanzini, aspects of holomorphic BF theory are presented. Holomorphic BF theory on a four dimensional Kahler manifold is discussed from page 5, and on page 8 the ...
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1answer
178 views

Map from local systems to holomorphic line bundles on a curve

Let $X$ be a Riemann surface of genus $g > 0$. Let $S$ denote the set of local systems (locally constant sheaves) on $X$ with fiber $\mathbb{C}$. $S$ is in natural bijection with $H^1(X, \underline{...
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0answers
325 views

A struggle with jets and Grothendieck vs Ehresmann connections

Let $X\to Y$ be a $C^\infty$ submersion. Consider the following two sheaves. The sheaf on $Y$ comprised of jets of sections of $X\to Y$. The sheaf on $X$ given by the quotient of $\Delta_{X/Y}^{-1}C^\...
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0answers
33 views

One parameter change of a section of $T^*M \otimes End(TM)$ on an affinely flat manifold

Let $\nabla$ be a flat symmetric connection in the tangent bundle of a smooth manifold $M.$ Let $A$ be a global section of $T^*M \otimes End(TM).$ Let $\phi:B(0,1) \to M$ be a local affine chart on $...
6
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0answers
309 views

Is there an analog of the Levi–Civita connection for schemes?

Is there an analog of the Levi–Civita connection for schemes? There exists algebraic de Rham theory, $n$-forms on vector bundles (algebraically), and familiar constructions from differential geometry....
4
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1answer
201 views

A consequence of Ambrose-Singer theorem on holonomy

Consider $\nabla$ a connection in a vector bundle above a smooth manifold $M$.Consider a local frame $\sigma=(\sigma_1, \sigma_2,...,\sigma_m )$ on a contractible open set $U\subset M$ and calculate ...
2
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0answers
74 views

Poincaré connection encode torsion and curvature

I'm trying to understand something that is written in Baez & Wise paper "Teleparallel Gravity as a Higher Gauge Theory". In section 4, they discuss Poincaré connection and, first of all, split ...

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