[Paper Review] Algebraic curvature tensors whose skew-symmetric curvature operator has constant rank 2

Peter Gilkey, Tan Zhang|ArXiv.org|May 8, 2002

Tensor decomposition and applications6 references4 citations

TL;DR

This paper classifies algebraic curvature tensors on pseudo-Riemannian manifolds of signature $(p,q)$ with $q \geq 5$ whose skew-symmetric curvature operator has constant rank 2. It shows these tensors are geometrically realizable as hypersurfaces in flat spaces and fully characterizes the Ivanov-Petrova tensors of rank 2 via self-adjoint operators with spacelike-definite kernels, extending prior results from Riemannian and Lorentzian settings to higher signatures.

ABSTRACT

Let R be an algebraic curvature tensor for a non-degenerate inner product of signature(p,q) where q>4. If $π$ is a spacelike 2 plane, let $R(π)$ be the associated skew-symmetric curvature operator. We classify the algebraic curvature tensors so R(-) has constant rank 2 and show these are geometrically realizable by hypersurfaces in flat spaces. We also classify the Ivanov-Petrova algebraic curvature tensors of rank 2; these are the algebraic curvature tensors of constant rank 2 such that the complex Jordan normal form of R(-) is constant.

Motivation & Objective

• To classify algebraic curvature tensors whose skew-symmetric curvature operator has constant rank 2 on pseudo-Riemannian manifolds of signature $(p,q)$ with $q \geq 5$.
• To determine which of these curvature tensors are geometrically realizable as hypersurfaces in flat spaces.
• To fully classify Ivanov-Petrova algebraic curvature tensors of rank 2, defined by the constancy of the complex Jordan normal form of the curvature operator across all spacelike 2-planes.
• To extend previous results on rank 2 curvature tensors from Riemannian and Lorentzian settings to arbitrary signatures.
• To analyze the behavior of such curvature operators on both spacelike and timelike 2-planes, revealing non-constant rank behavior in certain cases.

Proposed method

• Define the skew-symmetric curvature operator $ R(\pi) $ for each oriented spacelike 2-plane $ \pi $ via the inner product and curvature tensor, ensuring independence from the choice of orthonormal basis.
• Use the algebraic structure of curvature tensors satisfying the first Bianchi identity, skew-symmetry, and symmetry properties to restrict the class of possible tensors.
• Characterize rank 2 curvature tensors via self-adjoint operators $ \phi $ on the tangent space such that $ R = \pm R_\phi $, where $ R_\phi(x,y)z = \langle \phi y, z \rangle \phi x - \langle \phi x, z \rangle \phi y $.
• Impose the admissibility condition: $ \phi $ self-adjoint and $ \ker(\phi) $ containing no spacelike vectors, ensuring the curvature operator has rank 2.
• Construct explicit geometric realizations by defining a metric $ g_\phi $ on a neighborhood of the origin in $ \mathbb{R}^{(p,q)} $ such that the curvature tensor matches $ R_\phi $, using the induced metric from the operator $ \phi $.
• Analyze the complex Jordan normal form of $ R(\pi) $ to classify Ivanov-Petrova tensors, showing that constancy of this form characterizes $ R = \pm R_\phi $ for admissible $ \phi $.

Experimental results

Research questions

• RQ1 Which algebraic curvature tensors of signature $(p,q)$ with $ q \geq 5 $ have a skew-symmetric curvature operator of constant rank 2?
• RQ2 Can all such rank 2 curvature tensors be geometrically realized as hypersurfaces in flat pseudo-Riemannian spaces?
• RQ3 What characterizes Ivanov-Petrova algebraic curvature tensors of rank 2 in higher signature settings?
• RQ4 How does the rank of the skew-symmetric curvature operator behave on timelike 2-planes when the rank is constant on spacelike 2-planes?
• RQ5 Why does the classification fail in the $ q = 4 $ case, and what are the structural differences in such curvature tensors?

Key findings

• All algebraic curvature tensors of rank 2 on a non-degenerate inner product space of signature $(p,q)$ with $ q \geq 5 $ are of the form $ R = \pm R_\phi $ for some admissible self-adjoint operator $ \phi $.
• Such curvature tensors are geometrically realizable as hypersurfaces in flat pseudo-Riemannian spaces, with the induced metric constructed explicitly via the operator $ \phi $.
• The Ivanov-Petrova tensors of rank 2 are exactly those curvature tensors $ R $ for which the complex Jordan normal form of $ R(\pi) $ is constant across all oriented spacelike 2-planes $ \pi $, and these are precisely the tensors $ R = \pm R_\phi $ with admissible $ \phi $.
• In the case $ q = 4 $, the classification fails: there exist rank 2 curvature tensors not of the form $ R_\phi $, such as $ T^{(0,4)} $, which arises from the Hodge star on $ \Lambda^4 $ and has range $ \pi^\perp $.
• For $ q = p+1 \geq 5 $, the tensor $ R_\phi $ may not be Ivanov-Petrova, as the Jordan structure varies: $ R_\phi(\pi)^2 = 0 $ for some $ \pi $, but not for others.
• When $ q = p-1 \geq 5 $, $ R_\phi $ is Ivanov-Petrova for spacelike planes but has non-constant rank on timelike 2-planes: rank 2 for $ \pi_i^- = \operatorname{Span}\{e_1^-, e_i^-\\
• )), and rank 0 for $ \pi_{q+1}^- = \operatorname{Span}\{e_1^-, e_{q+1}^-\