Vol. 87, No. 2, 1980

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On curvature operators of bounded rank

Jaak Vilms

Vol. 87 (1980), No. 2, 467–473
Abstract

A curvature operator, that is, a linear map R : Λ2V Λ2V , has bounded rank 2r if it maps simple bivectors into bivectors of rank 2r. It is shown here that this condition is equivalent to the following:

∑
R (xi1 ∧ y1)∧ ⋅⋅⋅∧ R(xir+1 ∧yr+1) = 0

for all x1,,xr+1, y1,,yr+1 in V , with the sum taken over all permutations (i1,,ir+1) of (1,2,3,,r + 1). An application to Riemannian geometry is given.

Mathematical Subject Classification 2000
Primary: 53B20
Milestones
Received: 17 November 1978
Revised: 10 July 1979
Published: 1 April 1980
Authors
Jaak Vilms