Geometry of manifolds with area metric: multi-metric backgrounds
  We construct the differential geometry of smooth manifolds equipped with an algebraic curvature map acting as an area measure. Area metric geometry provides a spacetime structure suitable for the discussion of gauge theories and strings, and is considerably more general than Lorentzian geometry. Our construction of geometrically relevant objects, such as an area metric compatible connection and derived tensors, makes essential use of a decomposition theorem due to Gilkey, whereby we generate the area metric from a finite collection of metrics. Employing curvature invariants for multi-metric backgrounds we devise a class of gravity theories with inherently stringy character, and discuss gauge matter actions. 
Frederic P. Schuller, Mattias N.R. Wohlfarth