Dominated splitting for exterior powers and singular hyperbolicity

We show that an invariant splitting for the tangent map to a smooth flow over a compact invariant subset is dominated if, and only if, the exterior power of the tangent map admits an invariant dominated splitting. For a C1C1 vector field X on a 3-manifold, we obtain singular-hyperbolicity using only the tangent map DX of X   and a family of indefinite and non-degenerate quadratic forms without using the associated flow XtXt and its derivative DXtDXt. As a consequence, we show the existence of adapted metrics for singular-hyperbolic sets for three-dimensional C1C1 vector fields.