Structures on manifolds
by
Ivan'shin Pyotr
Kazan State University, Russia

Poster

It is well known that there exists almost complex structure on any orientable 2-dimensional manifold M, almost double and almost dual structures on M which satisfiy conditions J²=-I, J²=0 , J²=I ; J provides any T_xM, x in M, by structure of 1-dimensional modulo over algebras C, R(\epsilon) or R(e). All of this structures are always integrable.

It is necessary to point out that there is a structure of manifold over R(\epsilon) or R(e) on M only if Euler characteristic of M is zero.

I look through structures J, st J² = \lambdaE. (\lambda is a smooth function on M)

Theorem 1: There is a structure of following type: J² = \lambdaE₂, \lambda in C ^\infty(M) (E₂-identity matrix) on any compact smooth 2-dimensional manifold M.

Theorem 2: The structure J on compact smooth manifold M can be given st for allx in M, J(x) =/= 0.

Theorem 3: The structure J always can be transformed to quasydiagonal type in some neighbourhood of point x, where J²=0, J =/= 0, \lambda in C ^\omega(U(x)) . We have the following canonical type

J' = æ ç è 0 1 - \lambda 0 ö ÷ ø

Theorem 4: The structure J, J²=\lambdaE_2n, \lambda in C^\infty(M) with n eigenvalues \surd{\lambda} and n eigenvalues -\surd{\lambda} can be transformed to the following canonical type of J (J²=0, J =/= 0, \lambda in C ^\omega(U(x)) ):

J' = æ ç è [0] En - \lambdaEn [0] ö ÷ ø

Date received: April 7, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-25.