CONNECTION-FREE DIFFERENTIAL GEOMETRY

Albert Nijenhuis

Key words: Differential invariants, derivation, vector 1--form,
Nijenhuis tensor.

1991 Math. Subject Classification: 53A55, 53C15, 17B66, 17B70.

Two figures (fig1.pcx, fig2.pcx) not visible in DVI form.

Abstract: Both vector 1-forms (tensor fields of type (1,1)) and
Lie brackets map (tangent) vectors to the same.  The
former do so in a manner that is linear over the ring of
functions; the latter are bilinear, but only over the
constants. By composing one Lie bracket and $m$ vector
1-forms (we consider $2\le m\le 4$) in all possible ways
many maps of vector fields are obtained, bilinear over
the constants.

We study linear combinations of these maps with
some special properties:

1.  Bilinearity over the functions. This yields
the F-N bracket as a differential invariant of pairs of
vector 1-forms, as well as an additional differential
invariant $J$ of triples of vector 1-forms; both are
tensors of type (1,2).

2. Linearity over the functions in one variable,
while the dependence on the other variable extends to an
action on all tensor fields, as a derivation of type
(0,1).

We give a complete analysis of all linear
relations between these differential invariants.  Most
derive from a known relation between F-N brackets, and
from the Leibniz rule for derivations. There is, however,
an additional relation for quadruples of vector 1-forms
that is independent of these.

The commutator of two derivations of type (0,1)
gives rise to another, of type (0,2). Its ``index''
(i.e., its action on functions) is expressed in terms of
F-N brackets and the $J$-invariant.

Several of the calculations are made simple by
studying an appropriate graph. Others are performed with
the use of a computer.