We study the properties of the n-volumic scalar curvature in this note. Lott-Sturm-Villani's curvature-dimension condition CD(κ,n) was showed to imply Gromov's n-volumic scalar curvature ≥nκ under an additional n-dimensional condition and we show the stability of n-volumic scalar curvature ≥κ with respect to smGH-convergence. Then we propose a new weighted scalar curvature on the weighted Riemannian manifold and show its properties.

We consider complex projective space with its Fubini–Study metric and the X-ray transform defined by integration over its geodesics. We identify the kernel of this transform acting on symmetric tensor fields.

Let M be a compact Hamiltonian T−space, with finite fixed point set MT . An equivariant class is determined by its restriction to MT , and to each fixed point p ∈ MT and generic component of the moment map, there corresponds a canonical class τp. For a special class of Hamiltonian T−spaces, the value τp,q of τp at a fixed point q can be determined through an iterated interpolation procedure, and we obtained a formula for τp,q as a sum over ascending chains from p to q. In general the number of such chains is huge, and the main result of this paper is a procedure to reduce the number of relevant chains, through a systematic degeneration of the interpolation direction. The resulting formula for τp,q resembles, via the localization formula, an integral over a space of chains, and we prove that, for complex Grassmannians, τp,q can indeed be expressed as the integral of an equivariant form over a smooth Schubert variety.

Given a hyperbolic surface with geodesic boundary S, the lengths of a maximal system of disjoint simple geodesic arcs on S that start and end at @S perpendicularly are coordinates on the Teichm¨uller space T (S). We express the Weil-Petersson Poisson structure of T (S) in this system of coordinates, and we prove that it limits pointwise to the piecewise-linear Poisson structure defined by Kontsevich on the arc complex of S. At the same time, we obtain a formula for the first-order variation of the distance between two closed geodesics under Fenchel-Nielsen deformation.

In G2 manifolds, 3-dimensional associative submanifolds (instantons) play a role similar to J-holomorphic curves in symplectic geometry. In [21], instantons in G2 manifolds were constructed from regular J-holomorphic curves in coassociative submanifolds. In this exposition paper, after reviewing the background of G2 geometry, we explain the main ingredients in the proofs in [21]. We also construct new examples of instantons.