Journal of Lie Theory 20 (2010), No. 2, 295--309
Copyright Heldermann Verlag 2010
Scalar Invariants on Special Spaces of Equiaffine Connections
Zdenek Dusek
Palacky University, Faculty of Science, tr. 17. listopadu 1192/12, 771 46 Olomouc, Czech Republic
dusek@prfnw.upol.cz
The only basic scalar invariant in the general equiaffine geometry is the determinant of the Ricci tensor. For special equiaffine geometries, more scalar invariants can emerge. In this paper, we first investigate invariants of torsion-less connections with constant Christoffel symbols in R2. For this aim, we calculate invariants of the corresponding representation of the group SL(2, R) on the space R6 of Christoffel symbols. As a result, we find three bi-quadratic polynomials forming a Hilbert basis of this representation. An interesting phenomenon (rational involutive maps of higher degree) appears during the calculation. We also study representation of SL(2, R) on the 9-dimensional space of special equiaffine connections in R3 and corresponding invariants.
Keywords: Equiaffine connection, representation of a Lie group, invariant function, Hilbert basis of invariants, involutive rational mapping.
MSC: 53A55, 53B05, 16R50
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