A NONCOMMUTATIVE EXTENSION OF GRAVITY

The commutative algebra of functions on a manifold is extended to a noncommutative algebra by considering its tensor product with the algebra of n x n complex matrices. Noncommutative geometry is used to formulate an extension of the Einstein-Hilbert action. The result is shown to be equivalent to the usual Kaluza-Klein theory with the manifold SU(n) as an internal space, in a truncated approximation.