On the Kolář connection

Włodzimierz M. Mikulski

Address: Institute of Mathematics, Jagiellonian University, S. Łojasiewicza 6, Cracow, Poland

E-mail: Wlodzimierz.Mikulski@im.uj.edu.pl

Abstract: Let $Y\rightarrow M$ be a fibred manifold with $m$-dimensional base and $n$-dimensional fibres and $E\rightarrow M$ be a vector bundle with the same base $M$ and with $n$-dimensional fibres (the same $n$). If $m\ge 2$ and $n\ge 3$, we classify all canonical constructions of a classical linear connection $A(\Gamma ,\Lambda ,\Phi ,\Delta )$ on $Y$ from a system $(\Gamma ,\Lambda ,\Phi ,\Delta )$ consisting of a general connection $\Gamma $ on $Y\rightarrow M$, a torsion free classical linear connection $\Lambda $ on $M$, a vertical parallelism $\Phi \colon Y\times _ME\rightarrow VY$ on $Y$ and a linear connection $\Delta $ on $E\rightarrow M$. An example of such $A(\Gamma ,\Lambda ,\Phi ,\Delta )$ is the connection $(\Gamma ,\Lambda ,\Phi ,\Delta )$ by I. Kolář.

AMSclassification: primary 53C05; secondary 58A32.

Keywords: general connection, linear connection, classical linear connection, vertical parallelism, natural operators.

DOI: 10.5817/AM2013-4-223