Curvature operators: pinching estimates and geometric examples

Abstract

of results1.1.The most frequently used and best motivated curvature assumptions in qualitative Riemannian geometry are bounds on the sectional curvatures of the Riemann tensor R. Associated with R is the so-called curvature operator R : A 2 Tp M -> A 2 Tp M (Tp M = V henceforth).The symmetries of R (not including the Bianchi identity) imply that R is a symmetric operator.Special assumptions on R also have geometric consequences, for example:(i) a compact, connected and oriented Riemannian manifold with positive curvature operator R has the rational homology of the sphere (this is a nice theorem of D. Meyer proved in [4]);(ii) the positivity of R implies that the Gauss-Bonnet integrand is positive (cf.[12]).In both cases positive sectional curvature is known not to be sufficient {cf.[5] and [10] for(ii)).1.2.The linear map R has more geometric invariants than just its spectrum, since the geometrically relevant orthogonal group which acts on all occurring tensor spaces is 0 (V)./\ For example the Bianchi identity for R does not make sense for the action of 0 (A V).Further 0 (V)-invariants are connected with the rank of the eigenvectors.Notice then the following geometric examples: ^.(i) the eigenvector associated with the largest eigenvalue of R for the standard complex projective space is the Kahler form;( 1 )

Locations

• Annales Scientifiques de l École Normale Supérieure - View - PDF
• French digital mathematics library (Numdam) - View - PDF