Symmetric curvature tensor

Abstract

Recently, we have used the symmetric bracket of vector fields, and developed the notion of the symmetric derivation. Using this machinery, we have defined the concept of symmetric curvature. This concept is natural and is related to the notions divergence and Laplacian of vector fields. This concept is also related to the derivations on the algebra of symmetric forms which has been discussed by the authors. We introduce a new class of geometric vector fields and prove some basic facts about them. We call these vector fields affinewise. By contraction of the symmetric curvature, we define two new curvatures which have direct relations to the notions of divergence, Laplacian, and the Ricci tensor.