Curvature on a Principal Bundle

This chapter examines curvature on a principal bundle. The curvature of a connection on a principal G-bundle is a g-valued 2-form that measures, in some sense, the deviation of the connection from the Maurer-Cartan connection on a product bundle. The Maurer-Cartan form Θ‎ on a Lie group G satisfies the Maurer-Cartan equation. Let M be a smooth manifold. The chapter then pulls the Maurer-Cartan equation back and uses Proposition 14.3 to get the Maurer-Cartan connection. It also considers the second structural equation; the first structural equation is discussed in a previous chapter. Finally, the chapter derives some properties of the curvature form.

On noncommutative Levi-Civita connections

We make some observations about Rosenberg’s Levi-Civita connections on noncommutative tori, noting the non-uniqueness of general torsion-free metric-compatible connections without prescribed connection operator for the inner *-derivations, the nontrivial curvature form of the inner *-derivations, and the validity of the Gauss–Bonnet theorem for two classes of nonconformal deformations of the flat metric on the noncommutative two-tori, including the case of noncommuting scalings along the principal directions of a two-torus.

Convergence results for two Kähler–Ricci flows

In this note, we provide some general discussion on the two main versions in the study of Kähler–Ricci flows over closed manifolds, aiming at smooth convergence to the corresponding Kähler–Einstein metrics with assumptions on the volume form and Ricci curvature form along the flow.

Riemannian foliations and the kernel of the basic Dirac operator

In this paper, in the special setting of a Riemannian foliation en- dowed with a bundle-like metric, we obtain conditions that force the vanishing of the kernel of the basic Dirac operator associated to the metric; this way we extend the traditional setting of Riemannian foli- ations with basic-harmonic mean curvature, where Bochner technique and vanishing results are known to work. Beside classical conditions concerning the positivity of some curvature terms we obtain new rela- tions between the mean curvature form and the kernel of the basic Dirac operator

Vertical Chern Type Classes on Complex Finsler Bundles

Vertical Chern Type Classes on Complex Finsler BundlesIn the present paper, we define vertical Chern type classes on complex Finsler bundles, as an extension of thev-cohomology groups theory on complex Finsler manifolds. These classes are introduced in a classical way by using closed differential forms with respect to the conjugated vertical differential in terms of the vertical curvature form of Chern-Finsler linear connection. Also, some invariance properties of these classes are studied.

THE CONSERVED CHARGES AND INTEGRABILITY OF THE CONFORMAL AFFINE TODA MODELS

We construct infinite sets of local conserved charges for the conformal affine Toda model. The technique involves the abelianization of the two-dimensional gauge potentials satisfying the zero-curvature form of the equations of motion. We find two infinite sets of chiral charges and apart from two lowest spin charges, all the remaining ones do not possess chiral densities. Charges of different chiralities Poisson commute among themselves. We discuss the algebraic properties of these charges and use the fundamental Poisson bracket relation to show that the charges conserved in time are in involution. Connections to other Toda models are established by taking particular limits.