Fields of tangent sets and Hofmann cones
The Bony-Brezis theorem states that a closed subset F of a differentiable manifold M is invariant under the flow associated with a locally Lipschitzian vector field A if and only if for every \(p\in F\) the tangent vector A(p) belongs to the subtangent space of F at p. In the present paper the author generalizes this result to Lipschitz fields of subsets of tangent vectors on M; applying this generalization in the Lie semigroup setting he obtains a number of remarkable results concerning Lie semialgebras (the author proposes the term 'Hofmann wedge' instead of semialgebra). We only cite the following: (Theorem 6.2) Let W be a generating semialgebra in a finite-dimensional real Lie algebra, and suppose that p is a \(C^ 1\)-point of W such that the characteristic function does not vanish at p. If (p) is an eigenvalue of multiplicity one for ad p then the tangent hyperplane \(T_ p\) of W at p is a subalgebra of L. (Theorem 10.1) If every point in a generating finite-dimensional real Lie semialgebra W is a \(C^ 1\)-point (or if every point of W is an \(E^ 1\)- point) then either W is an invariant cone or L is almost abelian. [The latter result generalizes earlier results by \textit{A. V. Levichev}].
scientific article; zbMATH DE number 4004500
scientific article