Geometric approach to the theory of photon manifolds in gravitational fields

Various geometric problems that arise in the context of studies of the effects of strong gravitational lensing in the vicinity of black holes and other ultracompact objects are discussed. To analyze existing issues, we develop and generalize the main methods and ideas of the theory of photon surfaces. In particular, we define and study the geometric and physical properties of fundamental photon submanifolds, which are a natural generalization of the photon sphere in the Schwarzschild metric to the case of an arbitrary stationary geometry. The new photon surfaces are defined as Lorentzian submanifolds for which the classical umbilical condition is restricted to a tangent bundle of the class of geodesics with a given impact parameter. These objects have a number of useful analytical characteristics and are suitable for the analysis and classification of shadows and relativistic images in spaces with non-integrable geodesic equations. We also analyze the relationship between a new type of photon manifolds and the existence of Killing tensor fields of the second rank.