Levi's forms of higher codimensional submanifolds

D'Agnolo, Andrea ; Zampieri, Giuseppe

Atti della Accademia nazionale dei Lincei. Rendiconti Lincei. Matematica e applicazioni, Série 9, Tome 2 (1991) no. 1, pp. 29-33

Résumé

Let \( X \cong C^{n} \), let \( M \) be a \( C^{2} \) hypersurface of \( X \), \( S \) be a \(C^{2} \) submanifold of \( M \). Denote by \( L_{M}\) the Levi form of \( M \) at \( z_{0} \in S \). In a previous paper [3] two numbers \( s^{\pm} (S,p)\), \( p \in (\dot{T}^{*}_{S}X)_{z_{0}} \) are defined; for \( S = M \) they are the numbers of positive and negative eigenvalues for \( L_{M} \). For \( S \subset M \), \( p \in S \times_{M} \dot{T}^{*}_{S}X) \), we show here that \( s^{\pm} (S, p) \) are still the numbers of positive and negative eigenvalues for \( L_{M} \) when restricted to \( T^{C}_{z_{0}}S \). Applications to the concentration in degree for microfunctions at the boundary are given.