A certain space-time metric and smooth general connections (Q1074123)

The notion of general connection was introduced by the author [Math. J. Okayama Univ. 9, 99-164 (1960; Zbl 0202.211)]. Recently he gave a construction of a space with general connections, which has points swallowing geodesics [ibid. 24, 157-165 (1982; Zbl 0502.53050)]. A set which swallows geodesics is called a black hole. In this paper the space-time metric \[ (*)\quad d\sigma^ 2=-(1-4m^ 2/r^ 2)dt^ 2+(2/r)dt dr+r^ 2(d\theta^ 2+\sin^ 2\theta d^ 2\phi) \] is considered. It is shown that this metric has the curve \(r=0\) in \(R\times R^ 3\) as a black hole for the system of visible geodesics. Then a general connection is found which is smooth on \(R\times R^ 3\) and has the same system of geodesics as the one of the metric (*).