On geodesic mappings of Riemannian spaces with cyclic Ricci tensor

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1 Annales Mathematicae et Informaticae 43 (2014) pp 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

2 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.

3 On geodesic mappings of Riemannian spaces with cyclic Ricci tensor 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+

4 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.

5 On geodesic mappings of Riemannian spaces with cyclic Ricci tensor 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), [3] M. C. Chaki - U. C. De, On pseudo symmetric spaces, Acta Math. Hung., 54 (1989), [4] N. Sz. Szinjukov, Geodezicseszkije otrobrazsenyija Rimanovih prosztransztv, Moscow, Nauka, (1979)