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In this paper we determine the connected component of the identity of the isotropy subgroups of a given action of SU(n) on a connected manifold whose first and second integral Pontrjagin classes are zero and whose dimension is less than n — 8n/3 — 1. 1. General introduction. Let SUdz) be a differentiable transformation group acting on a connected manifold Al with dim Al < « 822/3 2, and assume that the fitst two integral Pontrjagin classes of Al are zero. The purpose of this paper is to determine the orbit structure of Al under the action of SU(n). To be more precise, in Theorem 2.3 we determine those closed connected simple subgroups H oí SU(n) such that the first integral Pontrjagin class of S\j(n)/H, P.{SU(n)/H; Z), is zero; in Theorem 2.5 we determine those closed connected simple subgroups H oí S\J(n) such that Pj(SU(«)///; Z) = 0 = P^SUM/tf; Z); in Theorem 2.9 we determine the connected component of the identity of a regular isotropy subgroup of the given above actions of SU(n) on Al. In Theorem 3.1 we then show how the connected component of the identity of all isotropy subgroups of SU(») is completely determined by the connected component of the identity of the principal isotropy subgroup, and we give a list of such possible subgroups. Wu-Chung Hsiang and Wu-Yi Hsiang considered this problem in [4]. One of the key steps in their approach was their knowledge of large subgroups of SU(2i). This forced them to put a severe restriction on Al. Our approach is more general in nature, and allows us a much less restrictive dimension requirement. The author wishes to thank his thesis advisor Wu-Yi Hsiang for the encouragement and help he received while writing this paper. Some of these results were included in the author's Ph.D. thesis at Brown University. In this section we recall several well-known facts. Let G be a compact conReceived by the editors June 23, 1970 and, in revised form, July 6, 1973. AMS (MOS) subject classifications (1970). Primary 57E15; Secondary 57E25.

𝑆𝑈(𝑛) actions on differentiable manifolds with vanishing first and second integral Pontrjagin classes