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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composition of two tangent vector fields and connection [closed]

Let $X, Y$ be two tangent vector fields on a manifold. Consider their composition: $(X \circ Y)(f) = X(Y(f)) = X^i\frac{\partial Y(f)}{\partial u^i} = X^i\frac{\partial Y^j}{\partial u^i}\frac{\...
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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\...
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3 votes
1 answer
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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 ...
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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 ...
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2 votes
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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 (...
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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 ...
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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: ...
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Finding the (1,1) component of $e^{-\mathbb{A}^2}$ for $\mathbb{A}$ a superconnection

Let $E=E^+\oplus E^-$be a holomorphic superbundle over a compact Kahler manifold, and $v:E^+\oplus E^-$ an odd bundle map. Assume that both $E^+$ and $E^-$ are endowed with Hermitian metrics, and ...
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Is the cohomology $H^1(X, \mathcal{E}^\nabla)$ trivial, for the sheaf of constants of an algebraic connection $\nabla$?

Suppose that $\pi:X\to S$ is a flat morphism between Noetherian, integral schemes (of characteristic zero, if need be). Let $\mathcal{E}$ be a locally free sheaf on $X$, and $$\nabla:\mathcal{E} \to \...
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Exact sequence for low-degree terms of relative de Rham cohomology

Let $\varphi:X\to Y$ be a morphism of schemes, smooth of relative dimension 1. The de Rham cohomology $H^\bullet(X/Y)$ is the result of applying the derived functor $R^\bullet\varphi_*$ to the de Rham ...
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Multisymplectic connections and topological invariants

I was wondering if there exists any notion of multisymplectic connection that naturally generalizes symplectic (Fedosov) connections in symplectic geometry. From symplectic connections, it is well ...
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Covariant momenta associated to higher order Lagrangians

Let $\pi:Y\rightarrow X$ be a fibered manifold with fibered coordinates $(U,x^i,y^\rho)$ (whenever local calculations are needed) and $m$ dimensional base $X$ ($\dim X=m$). Suppose that $L\in\Omega^m_{...
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3 votes
1 answer
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Torsion free (1,0)-connections on the holomorphic tangent bundle?

Let $M$ be a complex manifold. Consider a connection $\nabla$ on the holomorphic tangent bundle $T^{1,0}M$. The torsion of $\nabla$ is defined as the torsion of the induced connection $D$ on the real ...
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Koszul exterior connections

Let $(E,M)$ be a vector bundle over a riemannian manifold $M$ which is a module for the exterior forms of $M$. I define a Koszul exterior connection as an operator $\nabla$ such that: $$ \nabla : E \...
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Conceptual definition of the extension of a connection to 1-forms

I have a question that arose while reading Milnor's "Characteristic Classes". I will use the notation from that book. Let $M$ be a smooth manifold and let $\zeta$ be a complex vector bundle ...
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Supercommutator and covariant derivation

Let $(E,\nabla)\to X$ be a vector bundle endowed with a connection and $f\in End(E)$ a bundle endomorphism. We can express the covariant derivative of $f$ using a commutator $$\nabla f=[\nabla,f].$$ ...
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Embedding of a bundle with connection into a bundle with flat connection?

I'm looking for a generalization of Nash's embedding theorem (for Riemannian manifolds) to vector bundles with a connection. Given a smooth manifold $M$ together with a vector bundle $V$ on $M$ ...
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How to calculate Gauss Manin connection?

If $f:X\rightarrow B$ is a holomorphic family of compact complex manifold. Fix a $k$, then all the $H^k(X_t,\mathbb{C})$ is the same with respect to $t$. Say take a $d$-closed form $\alpha\in H^k(X_t,...
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3 votes
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Change of two normal coordinates based on two nearby points?

Let $M$ be a manifold and $L(M)$ be the tangent frame bundle on $M$. Let $\Gamma$ be a linear connection on $L(M)$ which induces a covariant derivative $\nabla$ on $TM$. Let $p, q$ be two ...
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6 answers
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The purpose of connections in differential geometry [closed]

I am currently reading through differential geometry as a mathematics graduate. Can somebody give me a brief explainer on the purpose of connections? I could also use explainers on differential forms. ...
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Writing a Taylor series with covariant derivatives (connections)?

A connection of a vector bundle $E$ on a manifold $M$ is a map $d_E: \Omega^0(E) \to \Omega^1(E)$ that extends uniquely to a map $d_E: \Omega^p(E) \to \Omega^{p+1}(E)$ while satisfying $$ d_E(\omega \...
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5 votes
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Torsion-free Cartan connections

Let $M$ a differentiable manifold and $H\subset G$ a Lie group with a closed subgroup such that $G/H$ is connected. A $H\subset G$-Cartan connection on $M$ can be defined by A principal $G$-bundle on ...
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Existence of connection on the pushforward of a sheaf with connection

Let $X$ be a scheme and $\mathcal{F}$ an $\mathcal{O}_X$-module with a connection $\nabla$. If $f:X\to Y$ is a morphism of schemes, then we have a pushforward of $(\mathcal{F},\nabla)$ as a $\mathcal{...
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2 votes
1 answer
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If any two triangles of equal area can be mapped via affine maps, what can we say about the geometry?

This is a cross-post. Let $(M,g)$ be a two-dimensional compact surface, endowed with a Riemannian metric. Fix $s>0$, and suppose that for any two geodesic triangles $A,B$ having area $s$, there ...
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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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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 ...
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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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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\...
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(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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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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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 ...
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2 votes
1 answer
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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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10 votes
2 answers
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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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1 vote
1 answer
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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 ...
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2 votes
1 answer
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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 ...
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5 votes
1 answer
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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, ...
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1 vote
1 answer
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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 ...
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4 votes
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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 ...
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18 votes
2 answers
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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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1 vote
0 answers
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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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1 vote
1 answer
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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 ...
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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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42 votes
5 answers
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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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1 vote
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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))^...
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3 votes
2 answers
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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 $\nabla$. ...
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8 votes
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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 $SU(2)$...
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4 votes
4 answers
754 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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1 vote
1 answer
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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 ...
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4 votes
2 answers
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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 ...
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2 votes
1 answer
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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 ...
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