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Scalar invariants on special spaces of equiaffine connections. (English) Zbl 1206.53014

The author firstly investigates invariants of torsionless connections with constant Christoffel symbols in \(\mathbb{R}^2\). For this aim, he calculates invariants of the corresponding representations of the group \(\text{SL}(2,\mathbb{R})\) on the space \(\mathbb{R}^6\) of Christoffel symbols. As a result, he finds three bi-quadratic polynomials forming a Hilbert basis of this representation. An interesting phenomenon (rational involutive maps of higher degree) appears during the calculation. The author also studies representations of \(\text{SL}(2,\mathbb{R})\) on the 9-dimensional space of special equiaffine connections in \(\mathbb{R}^3\) and corresponding invariants.

MSC:

53A55 Differential invariants (local theory), geometric objects
53B05 Linear and affine connections
16R50 Other kinds of identities (generalized polynomial, rational, involution)
53A15 Affine differential geometry