On Ricci Curvatures of Hypersurfaces in Abstract Wiener Spaces

✍ Scribed by Setsuo Taniguchi

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 907 KB
• Volume: 136
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1996.0028

✦ Synopsis

Wiener space will be given to study how the signs of their Ricci curvatures varies. As a result, it will be concluded that the Ricci curvature is no longer a geometrical object in contrast with the curvatures of finite dimensional manifolds.

1996 Academic Press, Inc. H Â2], l # B*, where ( } , } ) stands for the natural pairing of B and its dual space B*, and we have used the standard identification of H* and H so that B*/H*= H/B. For a nondegenerate infinitely differentiable function G: B Ä R d in the sense of the Malliavin calculus and b # R d , one can define a submanifold M=[G=b] in B [2, 18]. The Ricci curvature of M has been introduced and investigated by Getzler, Airault, van Biesen, Kazumi, and Shigekawa [7,1,3,19,11]. In particular, the Weitzenbo ck formula was established on M. For Ricci curvatures, see Section 2. In the present paper, we investigate Ricci curvatures of hypersurfaces [F=a] in B; i.e., we study the case that d=1. We shall construct examples where the sign of Ricci curvature can be determined. From the examples we shall observe that the Ricci curvature of a submanifold of an abstract Wiener space is no longer a geometrical quantity.