Geodesic mappings of affine-connected and Riemannian spaces

Abstract

Ri k -0jr , + rkirj -0kr , -r irk , Ri, Wihk ~ Rihk ----n ' __ ~h Rik) "~-~1 (~hR[jk ] --nl_l(~h~[ji]__q.R[ki]))hwhere (~ is the Kronecker symbol, 0i -O/Oz i, [i,j] denotes an alternation without division. An equiafflne space is defined as A,, with Rij = Rji. The spaces where the conditions R~j k = 0 (Rij " 0) hold are called flat (Ricci-flat, respectively). The space A,, belongs to the class C r (A,, E C r) if F~j (x) E C *. Riemannian Spaces. In the Riemannian space V,,, determined by the symmetric and nondegenerate metric tensor gij, Christoffel symbols of types I and II are introduced by the formulas Fijk --1/2(O(igj)k -Okgik) and F~j --gha Fij=, where gii are elements of the inverse matrix to glj, (i, j) denotes a symmetrization without division. The signature of the metrics is assumed, in generul, to be arbitrary. Christoffel symbols of type II are the natural connection (the s connection) of Riemannian spaces, with respect to which the metric tensor is covariantly constant, i.e., gij,k = O. Hereafter "," denotes a covariant derivative If e = -1, then Kn is said to be an elliptical Kdhlerian space K~, if e = +1, then Kn is said to be a hyperbolic K~hlerian space K +, and if e = 0 and Rgll II = m >__ 2, Kn is said to be an m-parabolic Kdhlerian space K ~ The space K ~ is called the parabolic Kdhlerian space K ~ The spaces K + and K~-must be of even dimension. The spaces Kff were first considered by P. A. Shirokov [91], the spaces K + were considered by P. K. Rashevskii [58], and the spaces K ~ were studied by V. V. Vishnevskii [5]. In the investigations mentioned these spaces are referred to as A-spaces. Independently from P. A. Shirokov the spaces K n were studied by E. KLlaler [114]. In the international literature these spaces are preferably referred to as Kghlerian spaces (see, for example, [4, 62, 149, 150]).By the term almost germitian spaces we denote all Pdemarmian spaces where the structure Fib(x) exists for which the conditions F~F~ = -6~; Fi~gj) ~ = 0 hold. Investigations [4,113,149] were devoted to the

On equivalency of various geometric structures

“…We now state some results which will be used to show the coincidence of class 3 ad class 4 for the symmetry and recurrency condition. [31], [29], [30]) Every concircularly recurrent as well as projective recurrent manifold is necessarily a recurrent manifold with the same recurrence form.…”

“…studied the recurrent notion on the projective curvature tensor as well as concircular curvature tensor. However, all these notions are equivalent to the recurrent manifold ( [22], [29], [30], [31]). Recently Singh [50] studied the recurrent condition on the M-projective curvature tensor, but from our paper (see, Section 6) it follows that such notion is equivalent to the notion of recurrent manifold.…”

TWO REMARKS ON PQε-PROJECTIVITY OF RIEMANNIAN METRICS

“…g andḡ are projectively equivalent. The theory of projectively equivalent metrics has a very long tradition in differential geometry, see for example [5,6,8,10,13] and the references therein. EXAMPLE 3.…”

“…REMARK 1. PQ -projectivity of the Riemannian metrics is a special case of the socalled F-planar mappings introduced and investigated in [9], whose defining equation, i.e. equation (1) in [9] clearly generalises equation (2) above.…”

Normal forms for pseudo-Riemannian 2-dimensional metrics whose geodesic flows admit integrals quadratic in momenta

“…Investigation of geodesically equivalent metrics is a classical problem in differential geometry, see the surveys [3,33,37] or/and the introductions to [31,32,34]. In particular, normal forms for geodesically equivalent Riemannian 2-dimensional metrics were already constructed by Dini [17].…”