Annales Mathematicae et Informaticae 43 (2014) pp. 13 17 http://ami.ektf.hu On geodesic mappings of Riemannian spaces with cyclic Ricci tensor Sándor Bácsó a, Robert Tornai a, Zoltán Horváth b a University of Debrecen, Faculty of Informatics bacsos@unideb.hu, tornai.robert@inf.unideb.hu b Ferenc Rákóczi II. Transcarpathian Hungarian Institute zolee27@kmf.uz.ua Submitted May 4, 2012 Accepted March 25, 2014 Abstract An n-dimensional Riemannian space V n is called a Riemannian space with cyclic Ricci tensor [2, 3], if the Ricci tensor R ij satisfies the following condition R ij,k + R jk,i + R ki,j = 0, where R ij the Ricci tensor of V n, and the symbol, denotes the covariant derivation with respect to Levi-Civita connection of V n. In this paper we would like to treat some results in the subject of geodesic mappings of Riemannian space with cyclic Ricci tensor. Let V n = (M n, g ij) and V n = (M n, g ij ) be two Riemannian spaces on the underlying manifold M n. A mapping V n V n is called geodesic, if it maps an arbitrary geodesic curve of V n to a geodesic curve of V n.[4] At first we investigate the geodesic mappings of a Riemannian space with cyclic Ricci tensor into another Riemannian space with cyclic Ricci tensor. Finally we show that, the Riemannian - Einstein space with cyclic Ricci tensor admit only trivial geodesic mapping. Keywords: Riemannian spaces, geodesic mapping. MSC: 53B40. 13
14 S. Bácsó, R. Tornai, Z. Horváth 1. Introduction Let an n-dimensional V n Riemannian space be given with the fundamental tensor g ij (x). V n has the Riemannian curvature tensor Rjkl i in the following form: R h ijk(x) = j Γ h ik(x) + Γ α ik(x)γ h jα(x) k Γ h ij(x) Γ α ij(x)γ h kα(x), (1.1) where Γ i jk (x) are the coefficients of Levi-Civita connection of V n. The Ricci curvature tensor we obtain from the Riemannian curvature tensor using of the following transvection: R α jkα (x) = R jk(x) 1. Definition 1.1. [2, 3] A Riemannian space V n is called a Riemannian space with cyclic Ricci tensor, if the Ricci tensor of V n satisfies the following equation: R ij,k + R jk,i + R ki,j = 0, (1.2) where the symbol, means the covariant derivation with respect to Levi-Civita connection of V n. Definition 1.2. [4] Let two Riemannian spaces V n and V n be given on the underlying manifold M n. The maps: γ : V n V n is called geodesic (projective) mappings, if any geodesic curve of V n coincides with a geodesic curve of V n. It is wellknown, that the the geodesic curve x i (t) of V n is a result of the second order ordinary differential equations in a canonical parameter t: d 2 x i dt 2 + Γi αβ(x) dxα dt dx β dt = 0. (1.3) We need in the investigations the next: Theorem 1.3. [4] The maps: V n V n is geodesic if and only if exits a gradient vector field ψ i (x), which satisfies the following condition: and Γ i jk(x) = Γ i jk(x) + δ i jψ k (x) + δ i kψ j (x), (1.4) Definition 1.4. [1] A Riemannian space V n is called Einstein space, if exists a ρ(x) scalar function, which satisfies the equation: R ij = ρ(x)g ij (x). (1.5) 1 The Roman and Greek indices run over the range 1,..., n; the Roman indices are free but the Greek indices denote summation.
On geodesic mappings of Riemannian spaces with cyclic Ricci tensor 15 2. Geodesic mappings of Riemannian spaces with cyclic Ricci tensors It is easy to get the next equations [4]: where ψ ij = ψ i,j ψ i ψ j and R ij = R ij + (n 1)ψ ij, (2.1) R ij,k = R ij x k Γα ik(x)r αj Γ α jk(x)r αi, (2.2) where Γ α ik(x) are components of Levi-Civita connection if V n. At now we suppose, that V n in a Riemannian space with cyclic Ricci tensor, that is R ij,k + R jk,i + R ki,j = 0. (2.3) Using (2.2) we can rewrite (2.3) in the following form: R ij x k Γα ik(x)r αj Γ α jk(x)r αi + R jk x i Γ α ji(x)r αk Γ α ki(x)r αj + R ki x j Γ α kj(x)r αi Γ α ij(x)r αk = 0. From (1.4) and (2.1) we can compute: (2.4) (R ij + (n 1)ψ ij ) x k (Γ α ik(x) + ψ i (x)δ α k + ψ k (x)δ α i )(R αj + (n 1)ψ αj ) (Γ α jk(x) + ψ j (x)δ α k + ψ k (x)δ α j )(R αi + (n 1)ψ αi )+ (R jk + (n 1)ψ jk ) x i (Γ α ji(x) + ψ j (x)δ α i + ψ i (x)δ α j )(R αk + (n 1)ψ αk ) (Γ α ki(x) + ψ k (x)δ α i + ψ i (x)δ α k )(R αj + (n 1)ψ αj )+ (R ki + (n 1)ψ ki ) x j (Γ α kj(x) + ψ k (x)δ α j + ψ j (x)δ α k )(R αi + (n 1)ψ αi ) That is (Γ α ij(x) + ψ i (x)δ α j + ψ j (x)δ α i )(R αk + (n 1)ψ αk ) = 0. R ij x k + R jk x i + R ki x j Γ α ik (x)r αj Γ α jk (x)r αi+ Γ α ji (x)r αk Γ α ki (x)r αj+ Γ α kj (x)r αi Γ α ij (x)r R ij,k + R jk,i + R ki,j αk+
16 S. Bácsó, R. Tornai, Z. Horváth +(n 1) ψ ij x k (n 1)Γα ik(x)ψ αj ψ i (x)r kj (n 1)ψ i (x)ψ kj ψ k (x)r ij (n 1)ψ k (x)ψ ij (n 1)Γ α jk(x)ψ αi ψ j (x)r ki +(n 1) ψ jk x i (n 1)ψ j (x)ψ ki ψ k (x)r ji (n 1)ψ k (x)ψ ji + (n 1)Γ α ji(x)ψ αk ψ j (x)r ik (n 1)ψ j (x)ψ ik ψ i (x)r jk (n 1)ψ i (x)ψ jk (n 1)Γ α ki(x)ψ αj ψ k (x)r ij +(n 1) ψ ki x j (n 1)ψ k (x)ψ ij ψ i (x)r kj (n 1)ψ i (x)ψ kj + (n 1)Γα kj(x)ψ αi ψ k (x)r ji (n 1)ψ k (x)ψ ji ψ j (x)r ki (n 1)ψ j (x)ψ ki (n 1)Γ α ij(x)ψ αk ψ i (x)r jk (n 1)ψ i (x)ψ jk ψ j (x)r ik (n 1)ψ j (x)ψ ik = 0. If we suppose, that V n has cyclic Ricci tensor we have: ( ) ψij (n 1) x k Γα ik(x)ψ αj Γ α jk(x)ψ αi + That is ( ψjk +(n 1) x i Γ α ji(x)ψ αk Γ α ki(x)ψ αj ( ψki +(n 1) x j Γα kj(x)ψ αi Γ α ij(x)ψ αk 4ψ i (x)r jk 4ψ j (x)r ki 4ψ k (x)r ij ) + ) + (n 1)(4ψ i (x)ψ jk + 4ψ j (x)ψ ki + 4ψ k (x)ψ ij ) = 0. (n 1)(ψ ij,k + ψ jk,i + ψ ki,j ) 4(ψ i (x)r jk + ψ j (x)r ki + ψ k (x)r ij ) 4(n 1)(ψ i (x)ψ jk + ψ j (x)ψ ki + ψ k (x)ψ ij ) = 0. (2.5) Theorem 2.1. V n and V n Riemannian spaces with cyclic Ricci tensors have common geodesics, that is V n and V n have a geodesic mapping if and only if exists a ψ i (x) gradient vector, which satisfies the condition: (n 1)(ψ ij,k + ψ jk,i + ψ ki,j ) 4(ψ i (x)r jk + ψ j (x)r ki + ψ k (x)r ij ) 4(n 1)(ψ i (x)ψ jk + ψ j (x)ψ ki + ψ k (x)ψ ij ) = 0.
On geodesic mappings of Riemannian spaces with cyclic Ricci tensor 17 3. Consequences A) If ψ ij = 0, then R ij = R ij, and ψ i,j = ψ i ψ j, so we obtain: ψ i (x)r jk + ψ j (x)r ki + ψ k (x)r ij = 0. (3.1) B) If the V n is a Riemannian space with cyclic Ricci tensor and at the same time is a Einstein space, then we get ρψ i (x)g jk + ρψ j (x)g ki + ρψ k (x)g ij = 0 that is so It means nψ i (x) + 2ψ i (x) = 0, (3.2) (n + 2)ψ i (x) = 0. (3.3) Theorem 3.1. A Riemannian-Einstein space V n with cyclic Ricci tensor admits into V n with cyclic Ricci tensor only trivial (affin) geodesic mapping. References [1] A. L. Besse, Einstein manifolds, Springer-Verlag, (1987) [2] T. Q. Binh, On weakly symmetric Riemannian spaces, Publ. Math. Debrecen, 42/1-2 (1993), 103 107. [3] M. C. Chaki - U. C. De, On pseudo symmetric spaces, Acta Math. Hung., 54 (1989), 185 190. [4] N. Sz. Szinjukov, Geodezicseszkije otrobrazsenyija Rimanovih prosztransztv, Moscow, Nauka, (1979)