This Article extends the geometric analysis of slow invariant manifolds in explosive kinetics developed by Creta et al. to three-dimensional and higher systems. Invariant manifolds can be characterized by different families of Lyapunov-type numbers, based either on the relative growth of normal to tangential perturbations or on the deformation of m-dimensional volume elements (if the manifold is m-dimensional) and of the complementary (n - m)-elements in the normal orthogonal complement. The latter approach, based on elementary concepts of exterior algebra, is particularly simple because the evolution of the relevant volume elements can be related to suitable local stretching rates, and local analysis can be performed directly from the knowledge of the Jacobian matrix of the vector field. Several examples of bifurcations of the points-at-infinity, which modify the manifold structure, are discussed for 3-D models of exothermic reactions.