Linear transports along paths in vector bundles. IV. Consistency with   bundle metrics
  The problem for consistency between linear transports along paths and real bundle metrics in real vector bundles is stated. Necessary and/or sufficient conditions, as well as conditions for existence, for such consistency are derived. All metrics (resp. transports) consistent with a given transport (resp. metric) are explicitly obtained. The special case of linear transports, generated by derivations of tensor algebras, of vectors is considered. Analogous problems are investigated in complex vector bundles endowed with Hermitian metrics. 
Bozhidar Z. Iliev (Institute for Nuclear Research and Nuclear Energy,   Bulgarian Academy of Sciences, Sofia, Bulgaria)