A NONCOMMUTATIVE EXTENSION OF GRAVITY

The commutative algebra of functions on a manifold is extended to a no ncommutative algebra by considering its tensor product with the algebr a of n x n complex matrices. Noncommutative geometry is used to formul ate an extension of the Einstein-Hilbert action. The result is shown t o be equivalent to the usual Kaluza-Klein theory with the manifold SU( n) as an internal space, in a truncated approximation.