Celebratio Mathematica — Kobayashi

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incollection S. S. Chern, M. do Carmo, and S. Kobayashi: “Minimal submanifolds of a sphere with second fundamental form of constant length,” pp. 59–75 in Functional analysis and related fields: Proceedings of a conference in honor of Professor M. Stone (University of Chicago, May 1968). Edited by F. E. Browder. Springer (New York), 1970. MR 0273546 Zbl 0216.44001

Let \( M \) be an \( n \)-dimensional manifold which is minimally immersed in a unit sphere \( S^{n+p} \) of dimension \( n+p \). Let \( h \) be the second fundamental form of this immersion; it is a certain symmetric bilinear mapping \[ T_x\times T_x\to T_x^{\perp} \] for \( x\in M \), where \( T_x \) is the tangent space of \( M \) at \( x \) and \( T_x^{\perp} \) is the normal space to \( M \) at \( x \). We denote by \( S \) the square of the length of \( h \). It is known that if \( M \) is moreover compact, then \[ \int_M\Bigl(\Bigl(2-\frac{1}{p}\Bigr)S - n\Bigr)S\cdot *1\geq 0, \] where \( *1 \) denotes the volume element of \( M \) (Simons [1968]; a slightly more general formula will be proved in §§2 and 3). It follows that if \( S\leq n/(2-1/p) \) everywhere on \( M \), then either

\( S=0 \) (i.e., \( M \) is totally geodesic), or
\( S = n/(2-1/p) \).

The purpose of the present paper is to determine all minimal submanifolds \( M \) of \( S^{n+p} \) satisfying \( S = n/(2-1/p) \). The proof depends upon the results presented by the first named author in his lectures on minimal submanifolds in Berkeley in the Winter of 1968, in which an exposition of the work of Simons [1968] was made by the use of moving frames. To describe our result, we begin with examples of minimal submanifolds.

S. S. Chern	Related	Volume
Marshall Harvey Stone	Related	Volume
Manfredo P. do Carmo	Related
Felix E. Browder	Related