General analytic and infinitesimal deformations of immersions. II. (Q1278058)

A system of seven differential equations, called \textit{fundamental system of the theory of infinitesimal deformations of immersions} is considered, involving local expressions for the coframe \(\tau\), the connection forms \(\Phi\) and the torsion forms associated with an immersion \(f:X\longrightarrow \Pi\) of a manifold \(X\) into a flat Riemannian manifold \(\Pi\) of arbitrary signature. Results on existence and uniquenes of solutions of this system are proved. Conditions are given that ensure that infinitesimal deformations become infinitesimal bendings. In order to handle difficulties arising in the case of indefinite metrics on \(\Pi\), the principal normal space \(W\) of the immersion is introduced as the subspace spanned by all normal vectors \(\nu\) such that the corresponding second fundamental form \(\text{II}^\nu\) is \(\neq0\). Let \(\nu_1,\dots, \nu_q\) be a basis of \(W\). The type number of \(f\) is introduced as the maximum of dimensions \(t\) of the subspaces \(V\) of the tangent space such that the values \(\omega^{\nu_\kappa} (u_\lambda)\) of the Weingarten operators applied to a basis \(u_1,\dots,u_t\) of \(V\) are linearly independent. Additional results are derived assuming that the dimensions \(q\) and \(t\) are constant on \(X\). Part I, cf. ibid. 41, No. 9, 19--32 (1997; Zbl 0923.53020).