Riemannian foliations with Ehresmann connection

It is shown that the structural theory of Molino for Riemannian foliations on compact manifolds and complete Riemannian manifolds may be generalized to a Riemannian foliations with Ehresmann connection. Within this generalization there are no restrictions on the codimension of the foliation and on the dimension of the foliated manifold. For a Riemannian foliation (M,F) with Ehresmann connection it is proved that the closure of any leaf forms a minimal set, the family of all such closures forms a singular Riemannian foliation (M,F¯¯¯¯). It is shown that in M there exists a connected open dense F¯¯¯¯-saturated subset M0 such that the induced foliation (M0,F¯¯¯¯|M0) is formed by fibers of a locally trivial bundle over some smooth Hausdorff manifold. The equivalence of some properties of Riemannian foliations (M,F) with Ehresmann connection is proved. In particular, it is shown that the structural Lie algebra of (M,F) is equal to zero if and only if the leaf space of (M,F) is naturally endowed with a smooth orbifold structure. Constructed examples show that for foliations with transversally linear connection and for conformal foliations the similar statements are not true in general.

Limits of the trivial bundle on a curve

We attempt to describe the rank 2 vector bundles on a curve C which are specializations of the trivial bundle. We get a complete classifications when C is Brill-Noether generic, or when it is hyperelliptic; in both cases all limit vector bundles are decomposable. We give examples of indecomposable limit bundles for some special curves. Comment: Final version, published in Epiga

Cohomology of Flat Principal Bundles

We invoke the classical fact that the algebra of bi-invariant forms on a compact connected Lie group G is naturally isomorphic to the de Rham cohomology H*dR(G) itself. Then, we show that when a flat connection A exists on a principal G-bundle P, we may construct a homomorphism EA: H*dR(G)→H*dR(P), which eventually shows that the bundle satisfies a condition for the Leray–Hirsch theorem. A similar argument is shown to apply to its adjoint bundle. As a corollary, we show that that both the flat principal bundle and its adjoint bundle have the real coefficient cohomology isomorphic to that of the trivial bundle.

Moduli spaces of bundles on an abelian variety

Let [Formula: see text] be a complex abelian variety and [Formula: see text] a complex reductive affine algebraic group. We describe the connected component, containing the trivial bundle, of the moduli spaces of topologically trivial principal [Formula: see text]-bundles and [Formula: see text]-Higgs bundles on [Formula: see text]. We also describe the moduli spaces of [Formula: see text]-connections and the [Formula: see text]-character variety for [Formula: see text].

Affine analogues of the Sasaki-Shchepetilov connection

For two-dimensional manifold M with locally symmetric connection ∇ and with ∇-parallel volume element vol one can construct a flat connection on the vector bundle TM ⊕ E, where E is a trivial bundle. The metrizable case, when M is a Riemannian manifold of constant curvature, together with its higher dimension generalizations, was studied by A.V. Shchepetilov [J. Phys. A: 36 (2003), 3893-3898]. This paper deals with the case of non-metrizable locally symmetric connection. Two flat connections on TM ⊕ (ℝ × M) and two on TM ⊕ (ℝ2 × M) are constructed. It is shown that two of those connections – one from each pair – may be identified with the standard flat connection in ℝN, after suitable local affine embedding of (M,∇) into ℝN.

Some remarks on representations up to homotopy

Motivated by the study of the interrelation between functorial and algebraic quantum field theory (AQFT), we point out that on any locally trivial bundle of compact groups, representations up to homotopy are enough to separate points by means of the associated representations in cohomology. Furthermore, we observe that the derived representation category of any compact group is equivalent to the category of ordinary (finite-dimensional) representations of the group.

First and second cohomology group of a bu ndle

Let (E, π, M) be a vector bundle. We define two cohomology groups associated to π using the first and second order jet manifolds of this bundle. We prove that one of them is isomorphic with a Čech cohomology group of the base space. The particular case of trivial bundle is studied

Symplectomorphism groups and embeddings of balls into rational ruled 4-manifolds

Let Mμ0 denote S2×S2 endowed with a split symplectic form $\mu \sigma \oplus \sigma $ normalized so that μ≥1 and σ(S2)=1. Given a symplectic embedding $\iota :B_{c}\hookrightarrow M^0_{\mu }$ of the standard ball of capacity c∈(0,1) into Mμ0, consider the corresponding symplectic blow-up $\widetilde {M}^0_{\mu ,c}$. In this paper, we study the homotopy type of the symplectomorphism group ${\mathrm {Symp}}(\widetilde {M}^0_{\mu ,c})$ and that of the space $\Im {\mathrm {Emb}}(B_{c},M^0_{\mu })$ of unparametrized symplectic embeddings of Bc into Mμ0. Writing ℓ for the largest integer strictly smaller than μ, and λ∈(0,1] for the difference μ−ℓ, we show that the symplectomorphism group of a blow-up of ‘small’ capacity c<λ is homotopically equivalent to the stabilizer of a point in Symp(Mμ0), while that of a blow-up of ‘large’ capacity c≥λ is homotopically equivalent to the stabilizer of a point in the symplectomorphism group of a non-trivial bundle $\mathbb {C}P^2\#\,\overline {\mathbb {C}P^2}$ obtained by blowing down $\widetilde {M}^0_{\mu ,c}$. It follows that, for c<λ, the space $\Im {\mathrm {Emb}}(B_{c},M^0_{\mu })$ is homotopy equivalent to S2 ×S2, while, for c≥λ, it is not homotopy equivalent to any finite CW-complex. A similar result holds for symplectic ruled manifolds diffeomorphic to $\mathbb {C}P^2\#\,\overline {\mathbb {C}P^2}$. By contrast, we show that the embedding spaces $\Im {\mathrm {Emb}}(B_{c},\mathbb {C}P^{2})$ and $\Im {\mathrm {Emb}}(B_{c_{1}}\sqcup B_{c_{2}},\mathbb {C}P^{2})$, if non-empty, are always homotopy equivalent to the spaces of ordered configurations $F(\mathbb {C}P^{2},1)\simeq \mathbb {C}P^{2}$ and $F(\mathbb {C}P^{2},2)$. Our method relies on the theory of pseudo-holomorphic curves in 4 -manifolds, on the computation of Gromov invariants in rational 4 -manifolds, and on the inflation technique of Lalonde and McDuff.