arXiv subjects

Raina Ivanova

Publications and source records attributed to Raina Ivanova.

7 recordsLinked to original sources

arXiv2002

Jordan Szabo algebraic covariant derivative curvature tensors

We show that if $\nabla R$ is a Jordan Szabo algebraic covariant derivative curvature tensor on a vector space of signature (p,q), where q is odd and p is less than q or if q is congruent to 2 mod 4 and if p is less than q-1, then $\nabla R=0$. This algebraic result yields an elementary proof of the geometrical fact that any pointwise totally isotropic pseudo-Riemannian manifold with such a signature (p,q) is locally symmetric.

math.DG↗

arXiv2002

Higher order Jordan Osserman Pseudo-Riemannian manifolds

We study the higher order Jacobi operator in pseudo-Riemannian geometry. We exhibit a family of manifolds so that this operator has constant Jordan normal form on the Grassmannian of subspaces of signature (r,s) for certain values of (r,s). These pseudo-Riemannian manifolds are new and non-trivial examples of higher order Osserman manifolds.

math.DG↗

arXiv2002

Complex IP curvature tensors

Let M be a pseudo-Riemannian manifold with a pseudo-Hermitian complex structure $J$. We give necessary and sufficient conditions that the curvature operator $R(π)$ is complex linear when $π$ is a $J$ invariant real 2 plane. Under this assumption, we study when M is complex IP - i.e. the spectrum, or more generally the Jordan normal form, of $R(π)$ is constant on the Grassmannian of complex spacelike or timelike lines. Methods from algebraic topology are used to obtain restrictions on the spectrum of a complex IP algebraic curvature tensor.

math.DG↗

arXiv2002

The Geometry of the Skew-Symmetric Curvature Operator in the Complex Setting

We construct almost complex algebraic curvature tensors for pseudo Hermitian inner products whose skew-symmetric curvature operator has constant Jordan normal form on the set of non-degenerate complex lines.

math.DG↗

arXiv2002

Szabo Osserman IP Pseudo-Riemannian manifolds

We construct a family of pseudo-Riemannian manifolds so that the skew-symmetric curvature operator, the Jacobi operator, and the Szabo operator have constant eigenvalues on their domains of definition. This provides new and non-trivial examples of Osserman, Szabo, and IP manifolds. We also study when the associated Jordan normal form of these operators is constant.

math.DG↗

arXiv2002

Spacelike Jordan Osserman algebraic curvature tensors in the higher signature setting

Let $R$ be an algebraic curvature tensor on a vector space of signature $(p,q)$ defining a spacelike Jordan Osserman Jacobi operator $\JJ_R$. We show that the eigenvalues of $\JJ_R$ are real and that $\JJ_R$ is diagonalizable if $p<q$.

math.DG↗

arXiv2002

The Jordan normal form of higher order Osserman algebraic curvature tensors

We construct new examples of algebraic curvature tensors so that the Jordan normal form of the higher order Jacobi operator is constant on the Grassmannian of subspaces of type $(r,s)$ in a vector space of signature $(p,q)$. We then use these examples to establish some results concerning higher order Osserman and higher order Jordan Osserman algebraic curvature tensors.

math.DG↗