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We provide two closed-form geodesic formulas for a family of metrics onStiefel manifold, parameterized by two positive numbers, having both theembedded and canonical metrics as special cases. The closed-form formulas allowus to compute geodesics by matrix exponential in reduced dimension for low-rankmanifolds. Combining with the use of Fr{\'e}chet derivatives to compute thegradient of the square Frobenius distance between a geodesic ending point to agiven point on the manifold, we show the logarithm map and geodesic distancebetween two endpoints on the manifold could be computed by {\it minimizing}this square distance by a {\it trust-region} solver. This leads to a newframework to compute the geodesic distance for manifolds with known geodesicformula but no closed-form logarithm map. We show the approach works well forStiefel as well as flag manifolds. The logarithm map could be used to computethe Riemannian center of mass for these manifolds equipped with the abovemetrics. We also deduce simple trigonometric formulas for the Riemannianexponential and logarithm maps on the Grassmann manifold.

Closed-form geodesics and trust-region method to calculate Riemannian logarithms on Stiefel and its quotient manifolds