Killing tensors and warped product

Abstract

We present some examples of Killing tensors and give their geometric interpretation. We give new examples of non-compact complete and compact Riemannian manifolds whose Ricci tensor satisfies the condition ∇ X (X, X) = 2 n+2 Xτ g(X, X). 0. Introduction. Killing tensors are symmetric (0, 2) tensors on a Riemannian manifold (M, g) satisfying the condition (K) ∇ X (X, X) = 0 for all X ∈ X(M) or equivalently C X,Y,Z ∇ X (Y, Z) = 0 for all X, Y, Z ∈ X(M) where X(M) denotes the space of all local vector fields on M , C denotes the cyclic sum and ∇ denotes the Levi-Civita connection of (M, g). The condition (K) is a generalization of the condition ∇ = 0. Another generalization of this condition is ∇ X (Y, Z) = ∇ Y (X, Z), which gives the class of Codazzi tensors. The Codazzi tensors are quite frequently used in Riemannian geometry. For example the second fundamental form of any hypersurface immersed in a Euclidean space is a Codazzi tensor. On the other hand it is difficult to find general examples of Killing tensors in the literature. It is only known that a Ricci tensor of any naturally reductive homogeneous space (and more generally of any D'Atri space) has this property. The aim of the present paper is to show that Killing tensors appear quite naturally in Riemannian geometry. We prove that on every warped product M 0 × f 1 M 1 ×. .. × f k M k there exists a Killing tensor Φ(X, Y) = g(SX, Y) such that the functions λ 0 = µ ∈ R and λ i = µ + C i f 2 i for i > 1 are eigenfunctions of S for any µ ∈ R and any real constants C i ∈ R − {0}. Conversely, let Φ(X, Y) = g(SX, Y) be a Killing tensor with an integrable almost product structure given by its eigendistributions and with eigenvalues µ, λ 1 ,. .. , λ k such that µ ∈ R is constant and j>0 D j ⊂ ker dλ i (i.e. ∇λ i ∈

On some $${\mathcal {AC}}^{\perp }$$ AC ⊥ manifolds

“…(3.6) with A = −2, we obtain a solution f (t) = tanh t. The corresponding AC ⊥ manifold M = R + × f 2 S n is complete (see [7]) and diffeomorphic to R n+1 . It also admits a pair of Einstein-Weyl structures which are conformally Einstein.…”

“…It follows that in both cases considered above we get a compact AC ⊥ manifold diffeomorphic to S n+1 . If = 1 and A = −2, then one can easily check that f (t) = tanh t. In that case, we obtain a complete non-compact AC ⊥ manifold diffeomorphic to R n+1 (see [7]). If A > −2, then we can obtain a solution which tends to infinity which may not be defined on the whole of R + .…”

Bihermitian Gray surfaces

“…On a riemannian manifold a distribution Ᏸ ⊂ T M is called umbilical [Jelonek 2000] if ∇ X X |Ᏸ ⊥ = g(X, X )ξ for every X ∈ (Ᏸ), where X |Ᏸ ⊥ is the Ᏸ ⊥ component of X with respect to the orthogonal decomposition T M = Ᏸ⊕Ᏸ ⊥ . The vector field ξ is called the mean curvature normal of Ᏸ.…”

“…We shall find the conditions on f, h to obtain the warped product metric ‫ސރ(‬ 1 , g f ) × h ‫ސރ(‬ 1 , 4 can), where g f = dt 2 + f 2 (t)θ 2 is the metric on the first copy of ‫ސރ‬ 1 and can is the standard Fubini-Studi metric on the second copy of ‫ސރ‬ 1 . Then the Ricci tensor of (U, g h ) has two eigenvalues λ, µ corresponding to the eigendistributions D λ = Ᏸ, D µ = Ᏸ ⊥ which are given by the following formulas [Jelonek 2000;Madsen et al 1997]:…”

“…We shall find the conditions on f, h to obtain the metric on the whole of k . Then the Ricci tensor of (U, g h ) has two eigenvalues λ, µ corresponding to the eigendistributions D λ = Ᏸ, D µ = Ᏸ ⊥ which are given by the following formulas (see [Besse 1987;Jelonek 2002b;2000;Madsen et al 1997):…”