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].
321
questions
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Hermitic connections on complex line bundles with imaginary curvature form
It is a simple fact that if $L \to B$ is a complex line bundle endowed with an Hermitian product and a compatible connection $\nabla$, then the curvature $F_\nabla$ is imaginary (and so are the local ...
- 5,282
2
votes
1
answer
134
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Proof about affine connections
I'm reading Nomizu & Sasaki's "Affine Differential Geometry: Geometry of Affine Immersions" and I'm having some trouble with Proposition 1.4.
I have an immersed surface in $M \hookrightarrow \...
- 310
9
votes
1
answer
634
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Generalized Dirac operators
So far I met three definitions of the so called generalized Dirac operator(or Dirac type operators. Everything takes place over Riemannian manifols $M$ and we have smooth hermitian vector bundle $S \...
- 9,150
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0
answers
180
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Nonabelian Hodge structure for noncompact curves and Hodge structure on the fundamental group
Nonabelian Hodge theory, introduced by C. Simpson and others, may be interpreted as a description of the (real) Hodge structure on the fundamental group (say, of a compact curve) in terms of some ...
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95
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Invariant Lagrangians of a connection and its derivatives: how do they look like?
Let
$$
L=L(\Gamma,\partial\Gamma,\ldots,\partial^n\Gamma)
$$
be a Lagrangian depending on a linear symmetric connection $\Gamma$ on the tangent space of a manifold $M$ together with its derivatives up ...
- 2,517
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223
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Is a non-flat hermitian connection determined uniquely by its holonomy and curvature?
How do I prove that gauge-equivalence classes of $U(1)$ connections on a line bundle $L\to M$ are determined uniquely by pairs $(\alpha,F)$, where
$$\alpha\in\text{Hom}(\pi_1(M),U(1)),~~~~F\in \Omega^...
- 605
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2
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Principal bundle approach to general relativity
I am curious if there is any literature (texbooks, mainly, but articles will do too, though I don't have easy access to any paid journal) that deals with general relativity by using Ehresmann ...
- 2,612
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2
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A question about flat connection
Let $E\to X$ be a complex flat vector bundle, and say $\nabla_0$ and $\nabla_1$ are two flat connections on it. Let $p:X\times[0, 1]\to X$ denote the projection onto the first factor. Is there a way ...
- 1,087
4
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2
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481
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Holonomy of a Ricci-flat affine connection
There is some link between Ricci-flatness and reduction of holonomy. For example a Kahler manifold is Ricci-flat if and only if it has at most $SU(n)$ holonomy rather than $U(n)$, and it's apparently ...
- 61.6k
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Levelt-Turrittin Theorem over p-adics (or the monodromy theorem)
Let $V$ be a finite dimensional vector space over $\mathbb{C}((t))$. Let $D:V\rightarrow V$ be a differential operator; i.e., an additive $\mathbb{C}$-linear map satisfying
$$
D(a.v)=(t\frac{d}{dt}a)....
- 2,681
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1
answer
361
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Parallel Transport on Hypersurface Spinor Bundle
So this has been driving me up a wall. I'm trying to digest parts of the Parker & Taubes paper, "On Witten's Proof of the Positive Energy Theorem." Here's a link:
https://projecteuclid.org/...
- 777
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answer
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Transferring connection information to associated bundles and back
This might not be research level but I've tried more than once to ask about this in MSE and it got nowhere. So I thought It's fair to at least try.
At the risk of repeating well known stuff I tried ...
- 7,549
4
votes
1
answer
280
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Connection, compatible with type (1, 1) tensor field
I met with the following problem. Consider real manifold $M^{2n}$ with operator field $R$ (that is the tensor field of type $(1,1)$). We are to find a symmetric connection $\Gamma^k_{ij}$ such that ...
12
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907
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Aren't Riemannian geodesics also geodesics of the associated Cartan geometry?
I was inspired by R. W. Sharpe's book on doing differential geometry through Cartan connections. Unfortunately, the book is fairly thin in terms of specific examples in Riemannian geometry, so I ...
- 769
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votes
0
answers
141
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Growth of norm of curvature under direct sum or existence of universal connection
For a Hermitian vector bundle E over a Riemannian $(X,g)$ and a unitary connection A we may define the sup norm of the curvature by:
$$\|R_A\|=\sup \{\| tracefree (R_A (v))\|_{op} \mid v \in \Lambda ^...
- 479
24
votes
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Vector bundle $L$ admits connection if and only if degree of every direct summand of $L$ divisible by $\text{char}\,k$, intuition
Consider the following theorem of Atiyah.
Let $X$ be a connected smooth projective curve over an algebraically closed field $k$. Then a vector bundle $L$ on $X$ admits a connection if and only if the ...
user61522
3
votes
1
answer
183
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Are non-linear connections with linear holonomy, linear?
Let $\pi\colon TM\to M$ be the tangent bundle of a differentiable manifold, let $E=TM\backslash 0$ be the slit tangent bundle, and let $V_eE$ be the kernel of $\pi_*$ at $e\in E$. The set $VE=\cup_{e\...
- 1,341
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votes
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484
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How algebraic is the holonomy map?
Let $G$ be either a compact, simple, simply-connected Lie group, or a simply-connected complex reductive group (so either $SU(n)$ or $SL(n,\mathbb{C})$, for instance), or even complex affine. I don't ...
- 33.9k
4
votes
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385
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Symmetries of non-Riemannian curvature tensor
The curvature tensor, $R_{ab}{}^c{}_d$, can be obtained from a connection which not necessarily is a metric connection.
By construction it is antisymmetric in the first two indices, since roughly ...
- 690
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0
answers
395
views
residue formula for connections on curves
Let $X$ be a smooth, projective curve over a field $k$ (characteristic zero is enough for me) and $E$ a line bundle on $X$. Assume that $E$ is equipped with an integrable logarithmic connection $\...
- 9
3
votes
0
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437
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Connection and reduction of the structure group
I apologize for this question which is not really research-level, but I don't get any answer on mathStackExchange, and I asked a professor in my university... which couldn't find the solution.
I am ...
- 1,326
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3
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556
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Special connection of vector bundle over real manifold
Let $E \rightarrow M$ be a vector bundle over a smooth manifold $M$ and let $g$ be a bundle metric. Does there exists a conection (maybe unique) $\nabla$ which is compatible with $g$. By this I mean: ...
- 131
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140
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Does the monodromy of such VHS have to be trivial
Consider a variation of polarized Hodge structure on a punctured disk. Suppose that connection preserves Hodge filtration (which is much stronger, than Griffiths transversality). Moreover assume that ...
- 375
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votes
2
answers
399
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What is the space for the coefficients of the connection 1-form of a connection in a vector bundle?
Let $E\to X$ is a a (smooth real) vector bundle with structure group some Lie group $G$. Suppose we have a (linear) connection $\nabla$ on $E$.
Is it true that if $A$ is the connection 1-form of ...
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1
answer
1k
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Ricci Curvature and the Chern Class of the Levi-Civita
For a (compact) Kahler manifold $M$, the Ricci tensor is the symmetric $2$-form
$$
r(u,v) = \text{tr}\big( w \mapsto (D_wD_u - D_uD_w - D_{[u,w]})v\big).
$$
The Ricci curvature is the $2$-form
$$
r(...
2
votes
1
answer
652
views
Metric, torsion free connections on principal bundles
I hope this is not too elementary, but I have asked this question at the math.stack site, but I have obtained no answers.
Let $F(M)$ be the frame bundle of a $n$-dimensional differentiable manifold $...
- 3,064
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answer
1k
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Differential forms along the fiber
Let $E \to M$ be a smooth fiber bundle. Instead of differential forms defined on the whole tangent bundle $TE$ one could also consider forms on the vertical tangent bundle $VE$, i.e. forms defined on ...
- 5,472
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3
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Is there a way to define a Lie derivative of a connection?
I've been reading a little bit about the definition of symmetries on General Relativity, and they are related with the concept of Killing vector, i.e., vectors along which the Lie derivative of the ...
- 690
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1
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1k
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Wedge product of Endomorphism-Valued Forms
To define characteristic classes in smooth vector bundles $E\longrightarrow M$ there is a more or less standard procedure: to choose a connection $\nabla$ and to derive the curvature $\Omega$, which ...
- 2,071
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2
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1k
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Yang-Mills Functional and Energy
I have a question about the meaning of Yang-Mills Functional.
It is stated everywhere that the Yang-Mills Functional is a measure of energy. But the formal definition of the Yang-Mills Functional is:
...
- 2,071
31
votes
2
answers
2k
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Unifying Geometry for Characteristic Classes
When working with characteristic classes (more concretely Chern classes), one finds at least four essentially distinct approaches:
Axiomatic Approach. See, for instance, Vector Bundles and K-Theory, ...
- 2,071
3
votes
0
answers
235
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Parallel Ricci condition - Status report and bibliography
First I'd like to point out that I'm not a mathematician but a physicist. Dealing with a (hopefully) new affine theory of gravity we have find that the equation of motion are not the usual Einstein's ...
- 690
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votes
1
answer
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semi flat connections
Let $L\to V$ be complex line bundle and $F_{t}:V\to V$, $t\in [0,1]$, be a loop of diffeomorphisms, $F_0=F_1=$ identity.
For every $x\in V$, we get a loop $\gamma_x(t)=\{F_t(x)\}$ whose class in $\...
22
votes
3
answers
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Manifolds admitting flat connections
For each Riemannian manifold one can construct the Levi-Civita connection. While this connection is unique, we can call a (Riemannian) manifold flat if the Levi-Civita connection is flat. However when ...
- 9,150
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0
answers
273
views
Symplectic form on moduli space of connections
Let $M$ be the moduli space of flat $GL(n,\mathbb{C})$ connections on a compact oriented surface, and $\alpha$ the natural symplectic form on it.
Is there any known construction of a bundle with a ...
- 21
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votes
1
answer
615
views
The "Rolle theorem" for sections of a vector bundle
1) Assume that $E\to M$ is a smooth real vector bundle and $\nabla$ is a connection. (We do not assume any metric compatibility since we do not fix a metric on $E$). Assume that ...
- 312
4
votes
1
answer
354
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Analogy between connections and $\ell$-adic sheaves: what happens with the residue?
There are many analogies between $\ell$-adic sheaves on varieties over finite fields and vector bundles with connections on varieties over fields of characteristic zero. I would like to know what is ...
- 41
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votes
1
answer
482
views
reference for Levelt-Turritin
Can anybody recommend a good reference to learn the Levelt-Turritin decomposition theorem of formal connections? An intuitive description of what it says would also be very appreciated.
- 51
4
votes
1
answer
536
views
Are constant connection coefficients uniquely determined by the (1,3) curvature coefficients?
Suppose that on a certain coordinate system the coefficients $\Gamma^i_{jk}$, $i,j,k=1,\cdots, n$, of a linear connection are constant. We do not require compatibility with a metric, however I am ...
- 1,341
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votes
1
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78
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Decomposition of Lie subspaces
If $M=G/H$ is a reductive homogeneous space then we can write $\frak{g}=\frak{m}+\frak{h}$
where $[\frak{h}, \frak{m}]\subset \frak{m}$. Here $\frak{g}$ and $\frak{h}$ are the Lie algebras of $G$ and $...
- 1,368
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votes
2
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403
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A cohomology group which depends on the connection
Warning: I am not a differential geometer, so some of the following might not make sense.
Background:
Let $w: (T\Omega)^k \to \mathbb{R}$ be a $k$-tensor on $\Omega$, an open subset of $\mathbb{...
- 12k
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1
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175
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Decomposing connections on extensions of the frame bundle
I have posted this question on math.stackexchange, without success. I'll make it brief:
Let $E\rightarrow M$ be an orientable vector bundle of rank n equipped with some Riemannian metric, $P:=F_{SO(n)...
- 455
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1
answer
2k
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A 3-connected graph property by Tutte
Tutte （1961): A graph $G$ is $3$-connected if and only if there exists a sequence
$G_0, ...,G_n$ of graphs that have the following two properties
1) $G_0 = K_4$ and $G_n = G$
2) $G_{i+1}$ has an ...
user39815
4
votes
1
answer
145
views
connections on principal bundles over $S^1$
Suppose $G$ is a compact connected Lie group and $P$ is a $G$-bundle over $S^1$, $A$ is a connection. Then we can choose a frame such that $A = a d\theta$ where $a\in \mathfrak{g}$ is constant. My ...
- 43
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163
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a technical question on the definition of connections with regular singularities
Let $X$ be a quasi-projective variety over a field $k$ of characteristic zero. A good compactification of $X$ means a projective variety $\overline{X}$ containing $X$ as the complement of a simple ...
- 61
8
votes
1
answer
265
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regular singularities and comparison isomorphism
Let $X$ be a smooth variety over some field $k \subset \mathbb{C}$. By a theorem of Grothendieck, one has a canonical isomorphism of complex vector spaces
$$
\mathbb{H}^j(X, \Omega_{X/k}^\bullet) \...
- 81
6
votes
1
answer
2k
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Connections on a Lie Group
A Lie group $G$ can be considered as a reductive homogeneous space in at least two different ways; $G/\{e\}$ and $G\times G/G^*$. In the first case, the canonical connection associated with the ...
- 1,368
4
votes
3
answers
751
views
Two ways to differentiate a section of vector bundle
Let $\pi:E\rightarrow M$ be a vector bundle, and $D$ a connection on it. Suppose $\sigma_1,\sigma_2\in\Gamma(E)$, $p\in M$, $V\in T_pM$ such that $\sigma_1(p)=\sigma_2(p)$. Are the following two ...
- 41
2
votes
1
answer
101
views
Normalizing the value of a principal connection at a point
Let $\nabla$ be a symmetric, linear connection on a smooth manifold $X$.
If $p \in X$ is any point, on a normal chart for $\nabla$ around $p$ it holds:
$$ \Gamma_{ij}^k (p) = 0 \ , $$ where $\Gamma_{...
- 1,054
5
votes
1
answer
2k
views
how is the dual connection defined?
Let $E$ be a vector bundle (i.e. locally free $\mathcal{O}_X$-module) on some smooth algebraic variety $X$ and let $\nabla: E \to E \otimes \Omega^1_X$ be an integrable connection.
I have seen that $...
- 53