It is a prominent conjecture (relating Riemannian geometry and algebraic topology) that all simply-connected compact manifolds of special holonomy should be formal spaces, i.e., their rational homotopy type should be derivable from their rational cohomology algebra already -- an as prominent as particular property in rational homotopy theory. Special interest now lies on exceptional holonomy $G_2$ and $Spin(7)$. In this article we provide a method of how to confirm that the famous Joyce examples of holonomy $G_2$ indeed are formal spaces; we concretely exert this computation for one example which may serve as a blueprint for the remaining Joyce examples (potentially also of holonomy $Spin(7)$). These considerations are preceded by another result identifying the formality of manifolds admitting special structures: we prove the formality of nearly Kahler manifolds. A connection between these two results can be found in the fact that both special holonomy and nearly Kahler naturally generalize compact Kahler manifolds, whose formality is a classical and celebrated theorem by Deligne-Griffiths-Morgan-Sullivan.
For compact manifolds with infinite fundamental group we present sufficient topological or metric conditions ensuring the existence of two geometrically distinct closed geodesics. We also show how results about generic Riemannian metrics can be carried over to Finsler metrics.
A quasiclassical approximation is constructed to describe the eigenvalues of the magnetic Laplacian on a compact Riemannian manifold in the case when the magnetic field is not given by an exact 2-form. For this, the multidimensional WKB method in the form of Maslov canonical operator is applied. In this case, the canonical operator takes values in sections of a nontrivial line bundle. The constructed approximation is demonstrated for the Dirac magnetic monopole on the two-dimensional sphere.
We study the asymptotics of the number N(t) of geometrically distinct closed geodesics of a Riemannian or Finsler metric on a connected sum of two compact manifolds of dimension at least three with non-trivial fundamental groups and apply this result to the prime decomposition of a three-manifold. In particular we show that the function N(t) grows at least like the prime numbers on a compact 3-manifold with infinite fundamental group. It follows that a generic Riemannian metric on a compact 3-manifold has infinitely many geometrically distinct closed geodesics. We also consider the case of a connected sum of a compact manifold with positive first Betti number and a simply-connected manifold which is not homeomorphic to a sphere.
We demonstrate how by using the intersection theory to calculate the cohomology of $G_2$-manifolds constructed by using the generalized Kummer construction. For one example we find the generators of the rational cohomology ring and describe the product structure.
We construct a canonical basis of two-cycles, on a $K3$ surface, in which the intersection form takes the canonical form $2E_8(-1) oplus 3H$. The basic elements are realized by formal sums of smooth submanifolds.
