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2
votes
0answers
109 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 ...
4
votes
1answer
157 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 ...
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 ...
10
votes
3answers
787 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 ...
10
votes
2answers
870 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 ...
2
votes
0answers
123 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 ...
1
vote
0answers
118 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 ...
5
votes
1answer
226 views

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 ...
1
vote
0answers
79 views

reference for Levelt-Turritin

Can anybody recommend a good reference to learn the Level-Turritin decomposition theorem of formal connections? An intuitive description of what it says would also be very appreciated.
4
votes
1answer
360 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 ...
5
votes
0answers
383 views

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 ...
0
votes
1answer
61 views

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 ...
7
votes
2answers
275 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 ...
0
votes
1answer
88 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, ...
10
votes
1answer
596 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 ...
1
vote
2answers
310 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 ...
2
votes
1answer
119 views

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 ...
4
votes
1answer
114 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 ...
4
votes
0answers
93 views

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 ...
8
votes
1answer
195 views

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) ...
3
votes
3answers
245 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 ...
1
vote
1answer
81 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 ...
4
votes
1answer
307 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 ...
8
votes
6answers
2k 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 ...
5
votes
0answers
241 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 ...
2
votes
1answer
134 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 ...
1
vote
0answers
127 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 ...
4
votes
1answer
509 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 ...
2
votes
1answer
209 views

comparison theorem for connections with regular singularities

This is something I've never understood. Let me first recall the classical case, in which you start with a smooth algebraic variety $X$ over $\mathbb{C}$. One has the algebraic de Rham complex ...
0
votes
0answers
119 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 ...
4
votes
2answers
204 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 $\nabla N=0$?
24
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 ...
1
vote
0answers
220 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 ...
3
votes
2answers
192 views

Kernel of an integrable holomorphic dee-bar connection is a holomorphic vector bundle

I'm looking for references for the following fact, to pass on to a grad student. I think I can prove it, but it is a bit sloppy and I'd rather have something which is already written up. Let $X$ be a ...
1
vote
1answer
201 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 ...
4
votes
1answer
216 views

locally free sheaves and logarithmic connections

I know that, when $\mathcal{F}$ is a coherent sheaf on a smooth algebraic variety $X$ over a field $k$ equipped with a connection $$ \nabla: \mathcal{F} \to \mathcal{F} \otimes \Omega^1_X, $$ then ...
6
votes
1answer
169 views

Conjecture on homotopy groups of moduli space of self-dual connections

In the paper Stability in Yang-Mills Theories (1983), Taubes puts a (topological) bound on the Hessian of the YM-action on $S^4$. He consequently conjectured: "The inclusion ...
0
votes
1answer
184 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
1answer
200 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 $c_1(E)$ ...
2
votes
0answers
155 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 ...
1
vote
3answers
540 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.
2
votes
1answer
329 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
630 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
155 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
1k 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
193 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 $X$?
0
votes
1answer
198 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 ...
1
vote
1answer
233 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 ...
2
votes
1answer
333 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 ...
0
votes
1answer
167 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