  In this note we generalize the methods of [1][2][3] to 5-dimensional
Riemannian manifolds M. We study the relations between the geometry of M and
the number of solutions to a generalized Killing spinor equation obtained from
a 5-dimensional supergravity. The existence of 1 pair of solutions is related
to almost contact metric structures. We also discuss special cases related to
$M = S1 \times M4$, which leads to M being foliated by submanifolds with
special properties, such as Quaternion-Kahler. When there are 2 pairs of
solutions, the closure of the isometry sub-algebra generated by the solutions
requires M to be S3 or T3-fibration over a Riemann surface. 4 pairs of
solutions pin down the geometry of M to very few possibilities. Finally, we
propose a new supersymmetric theory for N = 1 vector multiplet on K-contact
manifold admitting solutions to the Killing spinor equation.

High Energy Physics - Theory