A novel algorithm called the Proper Generalized Decomposition (PGD) is widely used
by the engineering community to compute the solution of high dimensional problems. However,
it is well-known that the bottleneck of its practical implementation focuses on the computation of
the so-called best rank-one approximation. Motivated by this fact, we are going to discuss some of
the geometrical aspects of the best rank-one approximation procedure. More precisely, our main
result is to construct explicitly a vector field over a low-dimensional vector space and to prove
that we can identify its stationary points with the critical points of the best rank-one optimization
problem. To obtain this result, we endow the set of tensors with fixed rank-one with an explicit
geometric structure.
