We give a summary of known results on Matveev’s complexity of compact 3-manifolds. The only relevant new result is the classification of all closed orientable irreducible 3-manifolds of complexity 10.

In this paper we review some important results on the closed geodesics problem for compact Riemannian manifolds, as the Gromoll-Mayer Theorem, and discuss some extension of those results to the case of Finsler and semi Rimannian manifolds.

From an important construction of Calabi (see [Ca], [Wo]), it follows that the compact Riemannian flat manifolds with first Betti number zero are the building blocks for all compact Riemannian flat manifolds. It is, therefore, of interest to construct families of such objects. These are often called primitive manifolds. Hantzsche and Wendt (1935) constructed the only existing 3-dimensional comp...

We give a new, connected-sum-like construction of Riemannian metrics with special holonomy G2 on compact 7-manifolds. The construction is based on a gluing theorem for appropriate elliptic partial differential equations. As a prerequisite, we also obtain asymptotically cylindrical Riemannian manifolds with holonomy SU(3) building up on the work of Tian and Yau. Examples of new topological types...

We prove regularity for a class of boundary value problems for first order elliptic systems, with boundary conditions determined by spectral decompositions, under coefficient differentiability conditions weaker than previously known. We establish Fredholm properties for Dirac-type equations with these boundary conditions. Our results include sharp solvability criteria, over both compact and non...

We give a hyperkk ahler analogue of the classiication of compact homogeneous KK ahler manifolds. Namely we show that, for a compact semisimple Lie group G, the complete G-invariant hyperkk ahler manifolds which are also locally G c-homogeneous with respect to one of the complex structures are precisely the hy-perkk ahler metrics of Kronheimer, Biquard and Kovalev on coadjoint semisimple orbits ...

In this paper we construct, for n ≥ 2, arbitrarily large families of infinite towers of compact, orientable Riemannian n-manifolds which are isospectral but not isometric at each stage. In dimensions two and three, the towers produced consist of hyperbolic 2-manifolds and hyperbolic 3-manifolds, and in these cases we show that the isospectral towers do not arise from Sunada’s method.

The celebrated Kodaira theorem [6] says that a compact complex manifold is projective if and only if it admits a Kähler form whose cohomology class is integral. This suggests that Kähler geometry is an extension of projective geometry, obtained by relaxing the integrality condition on a Kähler class. This point of view, together with the many restrictive conditions on the topology of Kähler man...

T T T T T T T T T T T T T T T Abstract In this paper we continue an earlier study of ends non-compact manifolds. The over-arching goal is to investigate and obtain generalizations of Siebenmann's famous collaring theorem that may be applied to manifolds having non-stable fundamental group systems at infinity. In this paper we show that, for mani-folds with compact boundary, the condition of inw...

We construct natural selfmaps of compact cohomgeneity one manifolds with finite Weyl group and compute their degrees and Lefschetz numbers. On manifolds with simple cohomology rings this yields in certain cases relations between the order of the Weyl group and the Euler characteristic of a principal orbit. We apply our construction to the compact Lie group SU(3) where we realize all possible de...