 
Summary: THE RUELLE ROTATION OF KILLING VECTOR
FIELDS
BY
KONSTANTIN ATHANASSOPOULOS (Iraklion)
Abstract. We present an explicit formula for the Ruelle rotation of a non
singular Killing vector field of a closed, oriented, Riemannian 3manifold, with
respect to Riemannian volume.
Let M be a closed, oriented Riemannian 3manifold and X be a non
singular Killing vector field on M with trivial normal bundle. The plane
bundle E orthogonal to X is then spanned by two globally defined orthogo
nal unit vector fields Y and Z, such that {X(x), Y (x), Z(x)} is a positively
oriented basis of the tangent space at x M. Once we have chosen the unit
vector field Z orthogonal to X, there is only one choice of a unit vetor field
Y such that {X, Y, Z} is a positively oriented orthogonal frame on M. The
flow (t)tR of X is a oneparameter group of isometries of M, and thus
t(x)(Ex) = Et(x), for every t R and x M. The matrix of t(x)Ex
with respect to the bases {Y (x), Z(x)} and {Y (t(x)), Z(t(x))} is a rota
tion, denoted by f(t, x). The resulting function f : R × M SO(2, R) is a
smooth cocycle of the flow, by the chain rule, and can be lifted to a smooth
cocycle ~f : R × M R. From the ergodic theorem for isometric systems
