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].
320
questions
1
vote
0
answers
62
views
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 ...
- 405
2
votes
0
answers
70
views
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_{...
- 2,612
3
votes
1
answer
243
views
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 ...
- 1,382
0
votes
0
answers
164
views
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:
$$
\nabla : E \...
- 377
25
votes
2
answers
1k
views
Conceptual definition of the extension of a connection to 1-forms
I have a question that arose while reading Milnor's "Characteristic Classes". I will use the notation from that book.
Let $M$ be a smooth manifold and let $\zeta$ be a complex vector bundle ...
- 269
9
votes
2
answers
479
views
Embedding of a bundle with connection into a bundle with flat connection?
I'm looking for a generalization of Nash's embedding theorem (for Riemannian manifolds) to vector bundles with a connection.
Given a smooth manifold $M$ together with a vector bundle $V$ on $M$ ...
- 587
2
votes
0
answers
197
views
How to calculate Gauss Manin connection?
If $f:X\rightarrow B$ is a holomorphic family of compact complex manifold. Fix a $k$, then all the $H^k(X_t,\mathbb{C})$ is the same with respect to $t$. Say take a $d$-closed form $\alpha\in H^k(X_t,...
- 29
3
votes
0
answers
141
views
Change of two normal coordinates based on two nearby points?
Let $M$ be a manifold and $L(M)$ be the tangent frame bundle on $M$. Let $\Gamma$ be a linear connection on $L(M)$ which induces a covariant derivative $\nabla$ on $TM$.
Let $p, q$ be two ...
- 261
3
votes
6
answers
2k
views
The purpose of connections in differential geometry [closed]
I am currently reading through differential geometry as a mathematics graduate.
Can somebody give me a brief explainer on the purpose of connections?
I could also use explainers on differential forms. ...
3
votes
0
answers
456
views
Writing a Taylor series with covariant derivatives (connections)?
A connection of a vector bundle $E$ on a manifold $M$ is a map $d_E: \Omega^0(E) \to \Omega^1(E)$ that extends uniquely to a map $d_E: \Omega^p(E) \to \Omega^{p+1}(E)$ while satisfying
$$
d_E(\omega \...
- 1,641
7
votes
1
answer
356
views
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 ...
- 141
3
votes
0
answers
257
views
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{...
- 4,246
2
votes
1
answer
228
views
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 ...
- 6,621
5
votes
1
answer
910
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{...
- 247
3
votes
1
answer
425
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 ...
- 247
5
votes
1
answer
493
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, $...
- 10.3k
0
votes
0
answers
95
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\...
- 853
8
votes
0
answers
242
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 ...
- 732
1
vote
0
answers
281
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 $H_A$ ...
- 111
3
votes
1
answer
475
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 ...
- 363
2
votes
1
answer
146
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 \...
- 3,301
10
votes
2
answers
2k
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$....
- 3,301
1
vote
1
answer
147
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 ...
- 543
2
votes
1
answer
298
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 ...
- 5,505
5
votes
1
answer
365
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, ...
- 1,337
1
vote
1
answer
304
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 ...
- 93
4
votes
1
answer
349
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 ...
- 5,505
19
votes
2
answers
869
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 ...
- 2,612
1
vote
0
answers
521
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 ...
- 4,776
1
vote
1
answer
368
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 ...
- 11
3
votes
1
answer
115
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 $\...
- 1,944
42
votes
5
answers
8k
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 ...
- 2,011
1
vote
0
answers
86
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))^...
- 767
4
votes
2
answers
874
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$. ...
- 43
9
votes
0
answers
214
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)$...
- 33.9k
6
votes
4
answers
1k
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 ...
1
vote
1
answer
264
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 ...
- 1,185
4
votes
2
answers
800
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 ...
- 121
2
votes
1
answer
172
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 ...
- 2,169
4
votes
2
answers
850
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 ...
- 543
1
vote
0
answers
181
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 ...
- 325
3
votes
0
answers
140
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 ...
- 183
8
votes
2
answers
392
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 ...
- 17.5k
4
votes
1
answer
239
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 ...
- 51
6
votes
0
answers
265
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 ...
- 1,253
2
votes
1
answer
248
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
votes
0
answers
301
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 $(\theta,\,\phi,\,\psi)$ ...
1
vote
1
answer
795
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 ...
- 5,931
2
votes
1
answer
244
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 ...
- 767
6
votes
1
answer
337
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
- 3,850