Title: Decomposability preserving curvature operators with an application to Einstein manifolds
Abstract: In this paper we examine curvature operators that preserve decomposability. In particular, we prove that if at each point of an Einstein manifold <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the sectional curvature operator is nonsingular and preserves decomposability, and the sectional curvature is either nonnegative or nonpositive, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M"> <mml:semantics> <mml:mi>M</mml:mi> <mml:annotation encoding="application/x-tex">M</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a space of nonzero constant curvature.