  We present an analysis of the stimulated Raman adiabatic passage processes
based on the methods of differential geometry. The present work was inspired by
an excellent article by Bruce W. Shore et al. (R. G. Unanyan, B. W. Shore, and
K. Bergmann Phys. Rev. A \textbf{59}, 2910 (1999)). We demonstrate how a purely
geometric interpretation of the adiabatic passage in quantum tripod systems as
a Riemannian parallel transport of the dark state vector along the Bloch sphere
allows describing the evolution of the system for a given sequence of Stokes,
pump and control laser excitation pulses. In combination with the
Dykhne-Davis-Pechukas adiabaticity criterion and the minimax principle for
circles on a sphere, this approach allows obtaining the analytical form of the
optimal laser pulse sequences for a high fidelity tripod fractional STIRAP. In
contrast to the conventional STIRAP in $\Lambda$-systems, the Gaussian
approximations of the optimal laser pulse sequences allow reaching the
infidelity of $10^{-7}$ for the adiabaticity parameter of $300$ without
noticeable oscillatory or other detrimental effects on population transfer
accuracy.
