Vol. 42, No. 3, 1972

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Locally holomorphic sets and the Levi form

Louis Roberts Hunt

Vol. 42 (1972), No. 3, 681–688

Suppose we have a real k-dimensional 𝒞2 manifold M embedded in Cn. If M has a nondegenerate complex tangent bundle of positive rank at some point p M, then the vanishing or nonvanishing of the Levi form on M near p determines whether or not M is locally holomorphic at p. We show that if M is locally holomorphic at p, then the Levi form vanishes near p, the converse being a known result. In addition we prove a C R extendibility theorem for a certain case when M is 𝒞 and has a nonzero Levi form at p M.

Mathematical Subject Classification
Primary: 32F15
Received: 7 May 1971
Revised: 29 October 1971
Published: 1 September 1972
Louis Roberts Hunt