  This work concentrates on optimization on Riemannian manifolds. The
Limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithm is a commonly
used quasi-Newton method for numerical optimization in Euclidean spaces.
Riemannian LBFGS (RLBFGS) is an extension of this method to Riemannian
manifolds. RLBFGS involves computationally expensive vector transports as well
as unfolding recursions using adjoint vector transports. In this article, we
propose two mappings in the tangent space using the inverse second root and
Cholesky decomposition. These mappings make both vector transport and adjoint
vector transport identity and therefore isometric. Identity vector transport
makes RLBFGS less computationally expensive and its isometry is also very
useful in convergence analysis of RLBFGS. Moreover, under the proposed
mappings, the Riemannian metric reduces to Euclidean inner product, which is
much less computationally expensive. We focus on the Symmetric Positive
Definite (SPD) manifolds which are beneficial in various fields such as data
science and statistics. This work opens a research opportunity for extension of
the proposed mappings to other well-known manifolds.
