Linear and non-linear refractive indices in Riemannian and topological spaces

Abstract

The refractive index and curved space relation is formulated using the Riemann-Christoffel curvature tensor. As a consequence of the fourth rank tensor of the Riemann-Christoffel curvature tensor, we found that the refractive index should be a second rank tensor. The second rank tensor of the refractive index describes a linear optics. It implies naturally that the Riemann-Christoffel curvature tensor is related to the linear optics. In case of a non-linear optics, the refractive index is a sixth rank tensor, if susceptibility is a fourth rank tensor. The Riemann-Christoffel curvature tensor can be formulated in the non-linear optics but there exist a gradient of susceptibility term. In topological space, we see that the linear and non-linear refractive indices are related to the Euler-Poincare characteristic, $\chi(M)$. Because the Euler-Poincare characteristic is topological invariant then the linear and non-linear refractive indices are also topological invariants.