  The primary aim of this thesis is to investigate metrics which are induced by
a differential form and arise as a critical point of Hitchin's variational
principle. Firstly, we investigate metrics associated with the structure group
PSU(3) acting in its adjoint representation. We derive various obstructions to
the existence of a topological reduction to PSU(3). For compact manifolds, we
also find sufficient conditions if the PSU(3)-structure lifts to an
SU(3)-structure. We give a Riemannian characterisation of topological
PSU(3)-structures through an invariant spinor valued 1-form and show that the
PSU(3)-structure is integrable if and only if the spinor valued 1-form defines
a co-closed Rarita-Schwinger field. Moreover, we construct non-symmetric
(compact) examples. Secondly, we consider even or odd forms which can be
naturally interpreted as spinors for a spin structure on $T\oplus T^*$. As
such, the forms we consider induce a reduction from $Spin(7,7)$ to $G_2\times
G_2$. We give a topological classification of $G_2\times G_2$-structures. We
prove that the condition for being a critical point is equivalent to the
supersymmetry equations on spinors in supergravity theory of type IIA/B with
NS-NS background fields. Examples are systematically constructed by the device
of T-duality.
