Celebratio Mathematica

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.

Marshall Harvey Stone	Related	Volume
Manfredo P. do Carmo	Related
Shōshichi Kobayashi	Related	Volume
Felix E. Browder	Related

incollection S.-S. Chern: “Circle bundles,” pp. 114–131 in Geometry and topology: Proceedings of the third Latin American school of mathematics (Rio de Janeiro, July 1976). Edited by J. Palis and M. P. Do Carmo. Lecture Notes in Mathematics 597. Springer (Berlin), 1977. MR 0461353 Zbl 0356.55005

Manfredo P. do Carmo	Related
Jacob Palis, Jr.	Related

incollection S. S. Chern: “An introduction to Dupin submanifolds,” pp. 95–102 in Differential geometry: A symposium in honour of Manfredo do Carmo (Rio de Janeiro, 1988). Edited by H. B. L. Lawson and K. Tenenblat. Pitman Monographs and Surveys in Pure and Applied Mathematics 52. Longman Scientific & Technical (Harlow, UK), 1991. MR 1173035 Zbl 0721.53052

Manfredo P. do Carmo	Related
H. Blaine Lawson, Jr.	Related
Keti Tenenblat	Related

incollection M. P. do Carmo: “S. S. Chern: Mathematical influences and reminiscences,” pp. 179–183 in S. S. Chern: A great geometer of the twentieth century. Edited by S.-T. Yau. Monographs in geometry and topology. International Press (Hong Kong), 1992. MR 1201358

Manfredo P. do Carmo	Related
Shing-Tung Yau	Related

S.-T. Yau, F. E. P. Hirzebruch, M. Atiyah, M. do Carmo, R. E. Greene, W.-l. Huang, K. Reich, L. Ji, J. Li, E. Kuh, Y. R. Shen, I. M. Singer, A. Weinstein, and J. A. Wolf: “Shiing-Shen Chern (1911–2004),” Notices Am. Math. Soc. 58 : 9 (October 2011), pp. 1226–1249. Hirzebruch’s contribution to this article was also published in Frontiers of mathematical sciences (2011). MR 2868099 Zbl 1228.01052 article

F. Hirzebruch	Related	Volume
Joseph Albert Wolf	Related
M. F. Atiyah	Related	Volume
Wen-ling Huang	Related
Karin Reich	Related
Ernest Kuh	Related
Lizhen Ji	Related
Robert Everist Greene	Related
Manfredo P. do Carmo	Related
Jun Li	Related
Y. Ron Shen	Related
Shing-Tung Yau	Related
Alan Weinstein	Related
Isadore M. Singer	Related	Volume

@article {key2868099m,
	AUTHOR = {Yau, Shing-Tung and Hirzebruch, Friedrich
		 Ernst Peter and Atiyah, Michael and
		 do Carmo, Manfredo and Greene, Robert
		 E. and Huang, Wen-ling and Reich, Karin
		 and Ji, Lizhen and Li, Jun and Kuh,
		 Ernest and Shen, Y. Ron and Singer,
		 Isadore M. and Weinstein, Alan and Wolf,
		 Joseph A.},
	TITLE = {Shiing-{S}hen {C}hern (1911--2004)},
	JOURNAL = {Notices Am. Math. Soc.},
	FJOURNAL = {Notices of the American Mathematical
		 Society},
	VOLUME = {58},
	NUMBER = {9},
	MONTH = {October},
	YEAR = {2011},
	PAGES = {1226--1249},
	URL = {http://www.ams.org/notices/201109/rtx110901226p.pdf},
	NOTE = {Hirzebruch's contribution to this article
		 was also published in \textit{Frontiers
		 of mathematical sciences} (2011). MR:2868099.
		 Zbl:1228.01052.},
	ISSN = {0002-9920},
}

year	title	people

			clear

Shiing-Shen Chern

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