Fields of geometrical objects on multidimensional tangentially framed surfaces in \(P_ n\). (Russian. English summary) Zbl 0549.53012

Dombrovskij, R. F.

Itogi Nauki Tekh., Ser. Probl. Geom. 7, 153-171 (1975).

A surface \(M_{m,r}\) in n-dimensional projective space \(P_ n\) at each point of which there is an r-dimensional plane lying in the tangent plane T(M) is called an m-dimensional tangentially r-framed surface. Suppose that an r-framed surface in a first order frame is given by the system of equations \(\omega^{\alpha}_ 0=0\), \(\omega^{\alpha}_ i=\lambda^{\alpha}_{ik}\omega^ k_ 0\), \(\Delta H_ a^{\xi}=H^{\xi}_{ak}=\omega^ k_ 0\), where \(\Delta\lambda^{\alpha}_ k(=\omega^{\alpha}_ k)\) and \(\Delta H^{\xi}_ a\) are Pfaffian forms and \(a,b=l,...,r\), \(\xi =r+l,...,m\); \(i,j=1,2,...,m\); \(\alpha,\beta =m+l,...,n\). The author uses the objects \(\{N^ i_{\alpha}\}\), \(\{b_ i\}\), \(\{N^ i_{\alpha}N_{\alpha}\}\), defining a normal field of the first kind, a normal field of the second kind and the field of (n-m-1)-dimensional framed planes, which were introduced by N. M. Ostianu.
For the surfaces \(M_{m,r}\in P_ n\) he constructs a field of one- parameter families of tangentially framed (m-r)-planes, intrinsically connected with the surface, finds three one-parameter pencils of (r-l)- planes lying in a tangentially r-framed plane, and studies the focal images connected with \(M_{m,r}\).

MSC:

53A55	Differential invariants (local theory), geometric objects
53A20	Projective differential geometry

Keywords:

projective space; framed surface; normal field; focal images