The geometry of modified Riemannian extensions

Abstract

We show that every paracomplex space form is locally isometric to a modified Riemannian extension and give necessary and sufficient conditions so that a modified Riemannian extension is Einstein. We exhibit Riemannian extension Osserman manifolds of signature (3,3) whose Jacobi operators have non-trivial Jordan normal form and which are not nilpotent. We present new four dimensional results in Osserman geometry.

References in corpus (7)

• The causal boundary of wave-type spacetimes
• Non-Walker Self-Dual Neutral Einstein Four-Manifolds of Petrov Type III
• Connections with skew-symmetric Ricci tensor on surfaces
• The geometry of recursion operators
• Projectively flat surfaces, null parallel distributions, and conformally symmetric manifolds
• A spinor approach to Walker geometry
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Cited by in corpus (11)

• Non-Walker Self-Dual Neutral Einstein Four-Manifolds of Petrov Type III
• Noncompactness and maximum mobility of type III Ricci-flat self-dual neutral Walker four-manifolds
• Gauge theory on projective surfaces and anti-self-dual Einstein metrics in dimension four
• Half conformally flat gradient Ricci almost solitons
• The moduli space of Type~A surfaces with torsion and non-singular symmetric Ricci tensor
• Bach-flat isotropic gradient Ricci solitons
• Constructing Bach Flat Manifolds of signature $(2,2)$ using the modified Riemannian extension
• Affine projective Osserman structures
• Ricci flow on modified Riemann extensions
• Einstein Metrics, Projective Structures and the $SU(\infty )$ Toda Equation
• Null-projectability of Levi-Civita connections