On a compact oriented Finsler Manifold without boundary, every Cohomology class has a Unique harmonic representative. The Dimension of the Space of all harmonic forms of degree is the th Betti Number of the Manifold.
See also Betti Number, Cohomology, Dimension, Finsler Manifold
Chern, S.-S. ``Finsler Geometry is Just Riemannian Geometry without the Quadratic Restriction.''
Not. Amer. Math. Soc. 43, 959-963, 1996.