... olΓjk=12gi∂gij∂xk+∂gik∂x j −∂gjk∂xi,where gijdenotes a component of the inverse metric (i.e., gijgjk= δik), we find the relationsde j ds=Γijkdxkdsei, and d2xds2=d2xids2ei+dxidsdeids=d2xids2+Γijkdx j dsdxkdsei.By ... refractive index n and the space metric isdenoted by gij.We now define the medium-modified space metricgij= n2gijasc2dt2= n2ds2= n2gijdxidx j = gijdxidx j , and apply the ... −tBtAgijd2x j dt2+∂gij∂xk−12∂gjk∂xidx j dtdxkdtδxidtc2.We now note that the second term can be written as∂gij∂xk−12∂gjk∂xidx j dtdxkdt=12∂gij∂xk+∂gik∂x j −∂gjk∂xidx j dtdxkdt1.3....