Conservation laws in differential geometry of plane curves and for eikonal equation and inverse problems

We derive the new conservation laws for a set of arbitrary smooth plane curves. In these laws a solenoidal field is expressed in terms of the Frenet unit vectors or in terms of the curvature vectors. When curves are vector lines of an arbitrary smooth vector field, these laws have identical form in terms of this field or its field of directions. Also, a series of vector analysis formulas as differential identities relating the modulus and direction of a vector field is obtained. It is based on these general formulas, the conservation laws for the kinematic seismics (geometrical optics) for a scalar time field, i.e., for the solutions of the eikonal equation are found. Some other formulas relating the time field and a characteristic of a medium (refractive index) are also given. In particular, we present the formula for determining an integral characteristic of a medium in the inverse problem formulation. All the formulas obtained originate from studying the differential invariants of a Lie group (an extension of the group of conformal transformations) which is realized as the equivalence group admitted by the eikonal equation and some other equations of mathematical physics.