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Transports preserving the angle between two contravariant vector fields butchanging their lengths proportional to their own lengths are introduced as''conformal'' transports and investigated over spaces with contravariant andcovariant affine connections (whose components differ not only by sign) andmetrics. They are more general than the Fermi-Walker transports. In ananalogous way as in the case of Fermi-Walker transports a conformal covariantdifferential operator and its conformal derivative are defined and consideredover the above mentioned spaces. Different special types of conformaltransports are determined inducing also Fermi-Walker transports for orthogonalvector fields as special cases. Conditions under which the length of a non-nullcontravariant vector field could swing as a homogeneous harmonic oscillator areestablished. The results obtained regardless of any concrete field(gravitational) theory could have direct applications in such types oftheories.  PACS numbers: 04.90.+e; 04.50.+h; 12.10.Gq; 02.40.VhComment: 17 pages, LaTe

CONFORMAL DERIVATIVE AND CONFORMAL TRANSPORTS OVER $\bm{({\bar L}_n,g)}$-SPACES