Differential geometry in the large and compactification of higher‐dimensional gravity

Some well‐known results from differential geometry are applied to some of the major issues of compactification of higher‐dimensional gravity. The results apply both to the theories generally known as Kaluza–Klein and the recently more promising super string theories. These results are primarily due to Yano [K. Yano, I n t e g r a l F o r m u l a s i n D i f f e r e n t i a l G e o m e t r y (Marcel Dekker, New York, 1970); D i f f e r e n t i a l G e o m e t r y o n C o m p l e x a n d A l m o s t C o m p l e x M a n i f o l d s (Macmillian, New York, 1965)] and have profound implications for the Kaluza–Klein scenario with respect to the cosmological constant problem and the massless sector of the theory. While the results are well known in the mathematical literature, the present author has only seen a fragmentary account presented by a few physicists. The necessary introduction to complex manifolds is also provided including Kähler manifolds and their possible relevance to the problem of compactification. The Ricci tensor provides the central role in the discussion of metric isometries, holomorphy, and holonomy. The incumbent role of Calabi–Yau manifolds with Ricci flat curvature and SU(n) holonomy, which have been recently conjectured in regard to super string compactification, is also mentioned.