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.
Top