Topics in the Geometry of Special Riemannian Structures

Abstract

The thesis consists of two chapters. The first chapter is the paper named “Betti numbers of nearly G₂ and nearly Kähler 6-manifolds with Weyl curvature bounds” which is now in the journal Geometriae Dedicata. Here we use the Weitzenböck formulas to get information about the Betti numbers of compact nearly G₂ and compact nearly Kähler 6-manifolds. First, we establish estimates on two curvature-type self adjoint operators on particular spaces assuming bounds on the sectional curvature. Then using the Weitzenböck formulas on harmonic forms, we get results of the form: if certain lower bounds hold for these curvature operators then certain Betti numbers are zero. Finally, we combine both steps above to get sufficient conditions of vanishing of certain Betti numbers based on the bounds on the sectional curvature. The second chapter is the paper written with my supervisor Spiro Karigiannis named “A special class of k-harmonic maps inducing calibrated fibrations”, to appear in the journal Mathematical Research Letters. Here we consider two special classes of k-harmonic maps between Riemannian manifolds which are related to calibrated geometry, satisfying a first order fully nonlinear PDE. The first is a special type of weakly conformal map u:(Lᵏ,g)→(Mⁿ,h) where k≤n and α is a calibration k-form on M. Away from the critical set, the image is an α-calibrated submanifold of M. These were previously studied by Cheng–Karigiannis–Madnick when α was associated to a vector cross product, but we clarify that such a restriction is unnecessary. The second, which is new, is a special type of weakly horizontally conformal map u:(Mⁿ,h)→(Lᵏ,g) where n≥k and α is a calibration (n-k)-form on M. Away from the critical set, the fibres u⁻¹{u(x)} are α-calibrated submanifolds of M. We also review some previously established analytic results for the first class; we exhibit some explicit noncompact examples of the second class, where (M,h) are the Bryant–Salamon manifolds with exceptional holonomy; we remark on the relevance of this new PDE to the Strominger–Yau–Zaslow conjecture for mirror symmetry in terms of special Lagrangian fibrations and to the G₂ version by Gukov–Yau–Zaslow in terms of coassociative fibrations; and we present several open questions for future study.