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0
votes
1answer
208 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 ...
4
votes
2answers
321 views

Chern class of a logarithmic connection

Let $X$ be a smooth complex projective algebraic variety and $E$ a line bundle on $X$. It is a classical result that if $E$ carries an integrable connection, then the first Chern class $c_1(E)$ ...
3
votes
0answers
170 views

Lefschetz hyperplane section theorem for connections

Let $X$ be a projective, smooth, algebraic variety over a subfield of the complex numbers, and let $Y \hookrightarrow X$ be a smooth hyperplane section of $X$. The classical Lefschetz theorem claims ...
3
votes
1answer
402 views

The (-)-Connection on a Lie Group

Is the geodesic exponential map for a Lie group with the (-)-connection a diffeomorphism? This connection is one of two flat connections introduced by Cartan and Schouten on a Lie group and has ...
8
votes
2answers
775 views

How many flat connections has a line bundle in algebraic geometry?

Suppose $X$ is a projective variety over $\mathbb C$. I am happy to entertain more or different adjectives — I'm not looking for the most general statement, but rather to understand when and how ...
0
votes
1answer
166 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 ...
17
votes
4answers
2k 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 ...
1
vote
0answers
199 views

flat connection

Suppose $p:P \to X$ is a projective bundle and $O(1)$ is the line bundle on $P$ restricting to $O(1)$ on each fibre. When is $p_*(O(1))$ flat on $X$?
4
votes
1answer
533 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 ...
1
vote
1answer
259 views

connections with regular singularities

Let $k$ be a field of characteristic zero, $X=\mathbb{G}_{m, k}=\mathrm{Spec}\ k[t, t^{-1}]$ the multiplicative group over k and $E=\mathcal{O}_X$ the trivial line bundle. Consider the connection ...
0
votes
1answer
213 views

regular singularities

Hi friends, Let me ask you about connexions having regular singularities. So imagine $X$ is some smooth algebraic variety over a subfield $k \subset \mathbb{C}$ and you have a locally free ...
9
votes
6answers
3k 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 ...
2
votes
1answer
386 views

cohomology of the Gauss-Manin connection

Let $U$ be a smooth algebraic variety defined over $k \hookrightarrow \mathbb{C}$. Let $\mathcal{E}$ be a locally free sheaf on $U$ equipped with an integrable connection $\nabla: \mathcal{E} \to ...
2
votes
2answers
256 views

decompose a connection

Is there any theorem which states any general linear connection can be decomposed into another linear connection plus the contortion tensor ? i didn't find any References
4
votes
3answers
358 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 ...
10
votes
2answers
995 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 ...
3
votes
1answer
742 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 ...
2
votes
3answers
640 views

Open problems in sub-Riemannian geometry

What are some open problems in sub-Riemannian geometry? I am interested especially in problems concerning connections and curvature, but any contribution is welcomed.
3
votes
2answers
1k 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 = ...
9
votes
1answer
631 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 ...
7
votes
4answers
353 views

Hamiltonians which commute both as operators and as connections

This is something which I suspect is written up in introductory books on mathematical physics if I knew where to look. Suppose I have some parameters $t_1$, ..., $t_k$ ranging over a neighborhood in ...
5
votes
0answers
183 views

Does every stack with a connection admit an atlas with a connection?

Dear all, Let $S$ be a scheme in characteristic $0$, and let $\mathscr{X}/S$ mean a crystal in Artin stacks over $S$ in the sense of this handout, page 4, Definition 0.5, where we replace the scheme ...
1
vote
2answers
443 views

complex vector bundles and curvature

Let us suppose that $X$, with a 2-form $\omega$. Suppose $J$ is an element of $su(2)$ such that $J^2=-e$ for $e$ the identity. Is there a necessary and sufficient condition on $\omega$ which will give ...
1
vote
2answers
1k 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 ...
3
votes
1answer
771 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 ...
0
votes
0answers
163 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 $x \in X$. ...
0
votes
1answer
333 views

A Follow-up About Connection Forms on Principal Bundles

In this question I asked about proving that a connection form $\alpha$ on a $\mathbb{C}^*$ bundle had to have $2\pi i(\alpha - \overline{\alpha})$ be exact. From the answer to that question I ...
3
votes
1answer
751 views

Principal bundle connections

It is my understanding that a connection form on a principal G-bundle over a manifold X is defined to be a Lie algebra-valued 1-form $\alpha$ which reproduces the Lie algebra generators of the ...
1
vote
2answers
322 views

Connections with compatible Hermitian products on complex line bundles

Let $X$ be a manifold, $L$ be a complex line bundle over $X$, and $L^{*}$ be the associated principal bundle. Suppose $\alpha$ is a connection form on $L^{*}$, with associated connection $D$ on $L$. ...
10
votes
1answer
669 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 ...
17
votes
3answers
3k 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 ...
5
votes
1answer
656 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 ...
6
votes
1answer
555 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 ...
11
votes
2answers
1k views

The algebraic Version of Riemann Hilbert Correspondence

It is well known that if I have a differentialable manifold (holomorphic maniford) $M$, then I have a functor from the categroy of vector bundles on $M$ with flat connections to the categroy of local ...
1
vote
2answers
951 views

tangent and cotangent bundle

Hi, I am reading "Introduction to symplectic topology" by McDuff and salamon. At some point I cant go further. My question is: Let $(M,g)$ be a Riemannian manifold and consider the cotangent bundle ...
6
votes
3answers
848 views

For quasi-coherent D-Modules

It is well-know that the category of coherent D-modules over a smooth algebraic $k$-scheme is a Tannakian category. So it is equivalent to the category of finite representations of some affine group ...
4
votes
1answer
395 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 ...
11
votes
3answers
3k views

Why is the integral of the second chern class an integer?

I'm currently reading the paper "Holonomy, the Quantum Adiabatic Theorem, and Berry's Phase" by Barry Simon. Imagine a vector bundle with a connection $\nabla$. For simplicity, we assume that this is ...
3
votes
1answer
572 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 ...
5
votes
1answer
608 views

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 ...
0
votes
0answers
248 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 ...
18
votes
4answers
2k 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 ...
2
votes
0answers
182 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 ...
3
votes
2answers
860 views

Why must a reducible flat SU(2)-connection over a homology sphere be trivial?

Let $M$ be a homology sphere. Suppose $P=M\times SU(2)$ is the trivial $SU(2)$ principal bundle. Let $R$ be all reducible connections on $P$. Here $A$ in $R$ is reducible if the gauge transformation ...
12
votes
3answers
489 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 ...
9
votes
5answers
2k views

pull-back connection

I have a question related to the definition of the pull-back connection, more specifically about its uniqueness or the canonical way to induce it. The definition that one finds in general goes ...
5
votes
0answers
458 views

Flat connections with logarithmic poles and p-adic parallel transport under choice of branch of logarithm

Consider the following simple situation: We work over the ring $R=\mathbb{Z}_p[[t]]$, over which we consider a rank $2$ free module $M$ with basis $(e,f)$. On $M$, we define a flat (topologically ...
1
vote
1answer
408 views

notion of connection on Torsors

Hi, Can anyone please enlighten me on the notion of connection for G-torsors? Edit (by WW) For some reason the OP decides to add clarification below as an answer rather than editing the question. ...
26
votes
5answers
3k 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 ...
4
votes
4answers
933 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 ...