Riemannsche Arealstrukturen und multiquadratische Formen

It is well known (and rather trivial to prove) that the square F² of a Finsler norm F:TM on a differentiable manifold M is differentiable at the zero section if and only if F is the norm function of a Riemannian metric. However, the corresponding question for a general p-areal on M is far less trivial to settle and leads to interesting algebraic and combinatorial problems concerning multiquadratic forms. For p=2, the results are closely related to known properties of the curvature tensor of a Riemannian metric.