All Questions
Tagged with dg.differential-geometry connections
201 questions
7
votes
1
answer
1k
views
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 ...
- 5,824
7
votes
3
answers
3k
views
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
3
votes
0
answers
238
views
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
8
votes
1
answer
1k
views
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 $\...
23
votes
3
answers
5k
views
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 ...
- 9,330
2
votes
0
answers
279
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
4
votes
1
answer
544
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 ...
- 1,371
7
votes
2
answers
421
views
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{...
- 12.1k
1
vote
1
answer
184
views
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)...
- 465
4
votes
3
answers
833
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
103
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_{...
- 1,064
0
votes
2
answers
435
views
Isomorphism of connections on a complex line bundle
Reading an article I faced with the following theorem, please give me a reference to a proof of the fact which is stated without any reference in the article. Is it a well-known fact?
Theorem. Let $E ...
- 1,329
2
votes
1
answer
171
views
Construction of a classifying map from a connection 1-form
From a connection 1-form on $M$, I can construct a parallel transport from which in turn I can construct a classifying map $M \to BG$.
Is there a construction of such a classifying map directly from a ...
- 3,880
20
votes
4
answers
3k
views
Can a manifold have a curvature-free connection that is not torsion-free?
Suppose I have a smooth manifold with a tangent bundle, and I have a connection. If this connection is curvature-free, is it guaranteed to be torsion-free? (I am not assuming a metric, just a finite-...
- 1,491
1
vote
0
answers
177
views
Obstruction to this gauge choice of the connection of a vector bundle
Let $M$ be a compact manifold with a nowhere-vanishing vector field $R$. Consider principal $G$-bundle $P$ over $M$, and $\mathcal{A}$ being the space of irreducible connections.
Let me denote a ...
- 857
9
votes
2
answers
2k
views
Does every manifold have a flat connection?
Suppose I have a manifold and a vector bundle over it, but not a connection or a metric. Can I always find a connection on it that has a Riemann curvature tensor that is identically zero? If so, can I ...
- 1,491
18
votes
3
answers
4k
views
Formal adjoint of the covariant derivative
Let $E \to M$ be a vector bundle over some Riemannian metric $(M, g)$ and endow it with some fibre metric. Assume that covariant derivative $\nabla$ is compatible with the metric.
It is essentially ...
- 5,824
0
votes
0
answers
248
views
General form of a metric affine connection with zero curvature
I have read somewhere that the general form of a metric affine connection whose Riemann curvature is zero is given by
$$\Gamma^{i}_{ij}=A^{i}_{\alpha}\partial_{j}A^{\alpha}_{k}$$
where $A^{i}_{\alpha}$...
- 573
4
votes
2
answers
554
views
Existence of connections making a bundle endomorphism parallel
Let $M$ be a manifold, $TM$ its tangent bundle, and $N:TM\to TM$ a vector bundle morphism. It is possible to find a torsionless linear connection $\nabla$ on $TM$ such that $\nabla N=0$?
- 153
1
vote
0
answers
354
views
Formula for the curvature of an induced connection
Let $\pi_P:P\to M$ be a principal $G$-bundle on a manifold $M$ endowed with a connection $A$ and $\pi_Q:Q\to M$ be a principal $H$-bundle on a manifold $M$. Let $f:P\to Q$ be a morphism of bundles ...
- 83
1
vote
1
answer
227
views
choices of connection in prequantization
In the definition of pre-quantization of representation $f\to \hat{f}$, (here $\hat{f}$ is Hermitian operator)of $C^{\infty}(M)$ on $L^2(M,L,\mu)$ where $\mu$ is Hermitian form, suppose that there ...
user21574
5
votes
0
answers
432
views
Anti-self-dual connections on CP^2
I'm learning Yang-Mills theory and its applications on 4-manifold.
I want to know that have someone computed all the anti-self-dual connections on principle
$SU(2)$ bundles over complex projective ...
- 303
1
vote
1
answer
602
views
Connections on tangent bundles and double tangent bundles
This can be viewed as a sequel to my previous question on double tangent bundle. Where I learned that the double tangent bundle $TTM$ is not natural diffeomorphic to $\oplus^3 TM$.
Recently, I also ...
- 1,271
1
vote
1
answer
261
views
Flat connection, finite-dimensional space of covariant constant one forms
hallo,
I have the following question: Let $U\subset \mathbb{R}^{n}$ be an open subset. Furthermore, let $\nabla$ be a flat connection on $U$ (not necessary Levi-Civita). How can one show that the ...
- 339
23
votes
4
answers
5k
views
Why is it important that partial derivatives commute?
I am asking this in the context of differential geometry (specifically Riemannian).
When the Levi-Civita Connection is defined, we require that the torsion tensor is 0, which in local coordinates ...
- 995
4
votes
1
answer
633
views
Why do we use the less simple convention for the definition of a vector bundle connection?
For a (smooth) vector bundle $F$ over a manifold $M$, one normally defines a connection to be a linear map
$$
\nabla:\Gamma^{\infty}(V) \to \Omega^1(M) \otimes \Gamma^{\infty}(V),
$$
satisfying $\...
- 637
19
votes
6
answers
9k
views
Tensor contraction and Covariant Derivative
What is the importance and intuition behind the the contraction operator on tensors (or the trace of a matrix, for that matter)?
In addition, I see that one of the requirements for a covariant ...
- 995
5
votes
3
answers
504
views
Connection Transformation Formula; Degree 3 Cech Cohomology
While reading through Brylinski, as in all of my posts, I am trying to understand the following equation:
$ g_* \tilde{\theta} = \tilde{\theta} - g^{-1} dg$
Setting
I have a principal $B$-bundle, $Q$...
- 1,611
13
votes
2
answers
3k
views
Intuition for Levi-Civita connection?
Levi-Civita connection is usually defined as the unique connection which is torsion free and preserves metric.
Question Is there some intuitively transparent constructive way to define it (or ...
- 24.9k
5
votes
1
answer
2k
views
1-jet bundle on vector bundle with metric connection
Background
I'm working to simplify the Lagrangian formalism of classical field theory for the situation of a vector bundle with a bundle metric and a metric connection. Particularly, I want to specify ...
3
votes
2
answers
2k
views
Interpretation of Curvature and Torsion
Dear all,
When dealing with General Relativity one uses the Levi-Civita connection with is torsion-free. Thus the commutator of the covariant derivatives yields
$[\nabla_\mu,\nabla_\nu]V^\rho = R_{\...
- 690
9
votes
1
answer
975
views
Is there a mathematical explanation for the Aharonov-Casher effect?
Recall that the Aharonov-Bohm effect can be interpreted mathematically as follows.
Consider an electromagnetic field A on some smooth manifold M, i.e., A is an element in the first differential ...
- 37.8k
0
votes
2
answers
3k
views
Line bundles with complex connection
Suppose that we have a complex manifold $X$, and a line bundle $L$ over $X$. It is known that the line bundles over $X$ are parametrized by their Chern class, the Chern class being the cohomology ...
- 1,025
7
votes
1
answer
2k
views
Terminology of "covariant derivative" and various "connections"
I apologize for the long question. My ultimate aim is to understand the existing terminology and find appropriate terminology for covariant derivatives on vector bundles which generalizes to "...
- 643
0
votes
0
answers
173
views
Analytic Characterization of Parallel Transport of Fundamental Groups
(Note that I've edited the main body of the question to make it clear for other readers.)
Fix a principal $G$-bundle $\rho: P \rightarrow X$ and fix a point $p \in P_x$ in the fiber above $x \in X$. ...
- 952
15
votes
1
answer
1k
views
Are bundle gerbes bundles of algebras?
The category of line bundles (possibly with connection)
on a smooth manifold M can be defined in two different ways:
The first definition uses transition functions that satisfy a cocycle condition
(...
- 37.8k
21
votes
3
answers
5k
views
Geometrical meaning of the Ricci Tensor and its Symmetry
Let $M$ be a smooth, pseudo-Riemannian manifold with $\dim(M) \ge 2.$ Let $\nabla$ be any affine connection on $M$. No reason for it to be the Levi-Civita connection. All we assume is that it has zero ...
- 581
7
votes
1
answer
1k
views
Symmetric Ricci Tensor
Let $M$ be a pseudo-Riemannian manifold. Assume that $M$ comes with a zero-torsion affine connexion $\nabla.$ There is no need for $\nabla$ to be the Levi-Civita connexion. Recall that the curvature ...
- 581
6
votes
1
answer
912
views
Almost Flat Connections On Principal G-Bundles
Let $E \rightarrow F$ be a principal $G$-bundle. Let $\alpha$ be a connection 1-form with values in the Lie algebra of $G$. Let $\omega$ denote the curvature 2-form of connection $\alpha$.
We know ...
- 439
4
votes
1
answer
620
views
"Nash Style" Embedding Theorem for Connections
The strong Whitney embedding theorem states that any smooth (Hausdorff and second-countable) manifold can be smoothly embedded in Euclidean space. John Nash went on to show that any Riemannian ...
- 3,399
3
votes
1
answer
955
views
Holonomy Groups and the Hopf Fibration
I am trying to understand holonomy groups at the moment and am focusing on the example of the Hopf fibration $SU_2 \to S^2$. Since $S^2$ is path connected we can talk about the holonomy group of a ...
- 3,399
5
votes
1
answer
1k
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Flat Principal Connections and Homotopy Groups?
I've come across a comment in a paper that hints at a relation between the set of (flat) connections for a principal bundle and the homotopy group of the base of the bundle. Can anyone tell me what ...
- 1,776
110
votes
6
answers
16k
views
When can a connection Induce a Riemannian metric for which it is the Levi-Civita connection?
As we all know, for a Riemannian manifold $(M,g)$, there exists a unique torsion-free connection $\nabla_g$, the Levi-Civita connection, that is compatible with the metric.
I was wondering if one can ...
- 3,399
1
vote
0
answers
325
views
Non-Existence of a Principal Connection for the Sphere over Projective Space?
As of the Wikipedia article on principal bundles connections:
Let $\pi: P \to M$ be a smooth principal bundle, a principal $G$-bundle over a smooth manifold $M$. Then a principal $G$-connection on $...
- 389
29
votes
4
answers
4k
views
Rolling without slipping interpretation of torsion
This is, in a sense, a follow up to this question.
Hehl and Obukhov try to give an intuitive description of torsion. I am confused about their description. I am looking at the following paragraph on ...
- 156k
2
votes
0
answers
224
views
Hermitian connections on real hypersurfaces of $\mathbb C^{n+1}$
I would like to find some non-trivial and "geometric" examples of hermitian connections (that is compatible with a give hermitian metric) on complex hermitian vector bundles over a smooth real ...
- 7,902
13
votes
3
answers
707
views
Can a homotopy inverse of the map from a Lie group to loops on its classifying space be given by holonomy?
Let $G$ be the compact Lie group $SO(n)$. There are some classical constructions of the classifying bundle of $G$ based upon on direct limits of Grassmann and Stiefel manifolds:
$$BG \simeq \...
- 53.3k
43
votes
5
answers
8k
views
A geometric interpretation of the Levi-Civita connection?
Let $M$ be a Riemannian manifold. There exists a unique torsion-free connection in the (co)tangent bundle of $M$ such that the metric of $M$ is covariantly constant. This connection is called the Levi-...
- 23.5k
4
votes
4
answers
1k
views
Proving the basic identity which implies the Chern-Weil theorem
If $E$ is a complex vector bundle over a manifold $M$ then one defines the space of vector valued $p$-differential forms on them as $\Omega^p(M,E) = \Gamma ( \wedge ^p (T^*M) \otimes E) $
The ...
- 3,541
10
votes
1
answer
2k
views
Global description of the Levi-Civita connection
I'm interested in finding a global (coordinate-free) description of the Levi-Civita connection on a (possibly infinite-dimensional) Riemannian manifold X.
I'm not looking for a description of this ...
- 8,803