Giambelli-type formula for subbundles of the tangent bundle

Abstract

Let us consider a generic n-dimensional subbundle V of the tangent bundle TM on some given manifold M. Given V one can define different degeneracy loci S_r(CV), r=(r_1<= r_2<= r_3<=...<=r_k) on M consisting of all points x in M for which the dimension of the subspace V^j(x) in TM(x) spanned by all length <= j commutators of vector fields tangent to V at x is less than or equal to r_j. We calculate 'explicitly' the cohomology classes dual to S_r(V) using determinantal formulas due to W.Fulton and the expression for the Chern classes of the associated bundle of free Lie algebras in terms of the Chern classes of V itself.