Geodesic Mappings and Their Generalizations

How many are affine connections with torsion

“…We substitute the arbitrary functions chosen above into the system (9) and transport them to the right-hand side, if necessary. We obtain a new system of equations of the form…”

“…For this purpose, we use the existence of the system of pre-semigeodesic coordinates. (see [9] for the definition and various applications of this concept and [2] for the alternative proof of the existence of such a system of coordinates).…”

Shells of monotone curves

“…where δ h i is the Kronecker symbol and the ψ i are differentiable functions [12], [15], page 143, [20].…”

“…Moreover, if R n is a (pseudo-)Riemannian space with respect to the metric tensor g, then the Lie group G is a group of isometries precisely if L ξ g = 0 (cf. [20], page 43, [15], page 100). A diffeomorphism ϕ is called a geodesic mapping if ϕ maps any geodesic onto a geodesic.…”

New integrability conditions of derivation equations in a subspace of asymmetric affine connection space

“…Sinyukov [14] introduced the concept of almost geodesic mappings between affine connected spaces without torsion. Mikeš [1], [5][6][7][8]13], [15], [18] gave some significant contributions to the study of geodesic and almost geodesic mappings of affine connected, Riemannian and Einstein spaces.…”