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].
289
questions
4
votes
1
answer
85
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 ...
- 939
3
votes
1
answer
106
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 ...
- 1,107
0
votes
0
answers
68
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 (...
- 939
10
votes
3
answers
451
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)...
- 4,255
3
votes
1
answer
75
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 ...
- 8,417
2
votes
0
answers
41
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 ...
- 2,097
6
votes
0
answers
166
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 $\...
- 55.4k
4
votes
1
answer
424
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 ...
- 263
8
votes
2
answers
274
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 ...
- 354
6
votes
1
answer
280
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-...
- 8,925
4
votes
1
answer
179
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\...
- 163
3
votes
1
answer
190
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 ...
- 171
3
votes
0
answers
207
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 ...
- 131
3
votes
0
answers
96
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 (...
- 1,052
5
votes
1
answer
335
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 ...
- 2,549
7
votes
1
answer
431
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: ...
- 535
2
votes
0
answers
52
views
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 ...
- 747
3
votes
0
answers
112
views
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
$$\nabla:\mathcal{E} \to \...
- 1,329
2
votes
0
answers
126
views
Exact sequence for low-degree terms of relative de Rham cohomology
Let $\varphi:X\to Y$ be a morphism of schemes, smooth of relative dimension 1. The de Rham cohomology $H^\bullet(X/Y)$ is the result of applying the derived functor $R^\bullet\varphi_*$ to the de Rham ...
- 1,329
1
vote
0
answers
55
views
Multisymplectic connections and topological invariants
I was wondering if there exists any notion of multisymplectic connection that naturally generalizes symplectic (Fedosov) connections in symplectic geometry.
From symplectic connections, it is well ...
- 405
2
votes
0
answers
69
views
Covariant momenta associated to higher order Lagrangians
Let $\pi:Y\rightarrow X$ be a fibered manifold with fibered coordinates $(U,x^i,y^\rho)$ (whenever local calculations are needed) and $m$ dimensional base $X$ ($\dim X=m$).
Suppose that $L\in\Omega^m_{...
- 2,097
3
votes
1
answer
183
views
Torsion free (1,0)-connections on the holomorphic tangent bundle?
Let $M$ be a complex manifold. Consider a connection $\nabla$ on the holomorphic tangent bundle $T^{1,0}M$. The torsion of $\nabla$ is defined as the torsion of the induced connection $D$ on the real ...
- 1,192
0
votes
0
answers
140
views
Koszul exterior connections
Let $(E,M)$ be a vector bundle over a riemannian manifold $M$ which is a module for the exterior forms of $M$. I define a Koszul exterior connection as an operator $\nabla$ such that:
$$
\nabla : E \...
- 357
25
votes
2
answers
935
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
416
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 $M$ ...
- 567
2
votes
0
answers
159
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
109
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 ...
- 201
3
votes
6
answers
1k
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. ...
2
votes
0
answers
337
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 \...
- 1,469
5
votes
0
answers
179
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 ...
- 121
2
votes
0
answers
183
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{...
- 3,984
2
votes
1
answer
224
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 ...
- 6,499
4
votes
1
answer
561
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{...
- 167
3
votes
1
answer
262
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 ...
- 167
5
votes
1
answer
484
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, $...
- 10.2k
0
votes
0
answers
90
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\...
- 595
7
votes
0
answers
206
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 ...
- 457
1
vote
0
answers
153
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
2
votes
1
answer
330
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 ...
- 203
2
votes
1
answer
137
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 \...
- 2,549
10
votes
2
answers
1k
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, $A(\mathcal{E}) = 0$....
- 2,549
1
vote
1
answer
133
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 ...
- 441
2
votes
1
answer
237
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 ...
- 5,112
5
votes
1
answer
302
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, ...
- 1,327
1
vote
1
answer
219
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
341
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 ...
- 5,112
19
votes
2
answers
807
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 ...
- 2,097
1
vote
0
answers
348
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 ...
- 4,516
1
vote
1
answer
319
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
112
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 $\...
- 1,480