Kernels of Algebraic Curvature Tensors of Symmetric and Skew-Symmetric Builds

Authors

  • Basia Klos University of Wisconsin-Madison

DOI:

https://doi.org/10.46787/pump.v6i0.3456

Keywords:

(canonical) algebraic curvature tensor; kernel of an algebraic curvature tensor

Abstract

The kernel of an algebraic curvature tensor is a fundamental subspace that can be used to distinguish between different algebraic curvature tensors. Kernels of algebraic curvature tensors built only of canonical algebraic curvature tensors of a single build have been studied in detail. We consider the kernel of an algebraic curvature tensor R that is a sum of canonical algebraic curvature tensors of symmetric and skew-symmetric build. An obvious way to ensure that the kernel of R is nontrivial is to choose the involved bilinear forms such that the intersection of their kernels is nontrivial. We present a construction wherein this intersection is trivial but the kernel of R is nontrivial. We also show how many bilinear forms satisfying certain conditions are needed in order for R to have a kernel of any allowable dimension.

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Published

2023-08-26

How to Cite

Klos, B. (2023). Kernels of Algebraic Curvature Tensors of Symmetric and Skew-Symmetric Builds. The PUMP Journal of Undergraduate Research, 6, 301–316. https://doi.org/10.46787/pump.v6i0.3456