Fig. 1

Example of \(Sym_{+}^{d}\) (d = 2) embedded in a 3D space \(\mathbb {R}^{3}\). O is the origin. P and Q are the manifold points, i.e., \(\mathbf {P}, \mathbf {Q} \in Sym_{+}^{d}\). \(\mathcal {T}_{\mathbf {P}}\) is the tangent space at P. \(\mathbf {\Delta } \in \mathcal {T}_{\mathbf {P}}\) is the tangent vector whose projected point on the manifold is Q. The geodesic ρ is the shortest curve between P and Q on the manifold