Beschreibung:
We review the geometry of second order linear operators acting on algebra of densities based on the work [15]. We consider the commutative algebra of densities on a manifold. This algebra is endowed with the canonical scalar product and we shall consider self-adjoint operators on this algebra. We naturally come to canonical pencils of operators passing through a given operator. There are singular values of weights for operators of a given order. These singular values lead to an interesting geometrical picture. In particular we come to operators which depend on classes of connections and we study a groupoid of connections such that orbits of this groupoid are these equivalence classes. Based on this point of view we analyze two examples: second order canonical operator in odd symplectic supermanifold appearing in Batalin-Vilkovisky geometry and Sturm-Lioville operator on the line related with classical constructions in projective geometry. We consider also the example of canonical second order semidensity in odd symplectic supermanifold. This semidensity has some resemblance to mean curvature in Riemannian geometry.