# Shoshichi Kobayashi

Shoshichi Kobayashi
Shōshichi Kobayashi in Berkeley
BornJanuary 4, 1932
Kōfu, Japan
DiedAugust 29, 2012 (aged 80)
Kofu, Yamanashi, Japan
NationalityJapanese
Known forKobayashi–Hitchin correspondence
Kobayashi metric
AwardsGeometry prize (1987)
Scientific career
FieldsMathematics
InstitutionsUniversity of California, Berkeley
Doctoral advisorCarl B. Allendoerfer
Doctoral studentsToshiki Mabuchi
Michael Minovitch
Burt Totaro

Shoshichi Kobayashi (小林 昭七, Kobayashi Shōshichi, born on January 4, 1932, in Kōfu, Japan, died on 29 August 2012)[1] was a Japanese mathematician. He was the eldest brother of electrical engineer and computer scientist Hisashi Kobayashi.[2] His research interests were in Riemannian and complex manifolds, transformation groups of geometric structures, and Lie algebras.

## Biography

Kobayashi graduated from the University of Tokyo in 1953. In 1956, he earned a Ph.D. from the University of Washington under Carl B. Allendoerfer. His dissertation was Theory of Connections.[3] He then spent two years at the Institute for Advanced Study and two years at MIT. He joined the faculty of the University of California, Berkeley in 1962 as an assistant professor, was awarded tenure the following year, and was promoted to full professor in 1966.

Kobayashi served as chairman of the Berkeley Mathematics Dept. for a three-year term from 1978 to 1981 and for the 1992 Fall semester. He chose early retirement under the VERIP plan in 1994.

The two-volume book Foundations of differential geometry (1963-1969), which he coauthored with Katsumi Nomizu, has been known for its wide influence.

## Technical contributions

As a consequence of the Gauss-Codazzi equations and the commutation formulas for covariant derivatives, James Simons discovered a formula for the Laplacian of the second fundamental form of a submanifold of a Riemannian manifold.[4] As a consequence, one can find a formula for the Laplacian of the norm-squared of the second fundamental form. This "Simons formula" simplifies significantly when the mean curvature of the submanifold is zero and when the Riemannian manifold has constant curvature. In this setting, Shiing-Shen Chern, Manfredo do Carmo, and Kobayashi studied the algebraic structure of the zeroth-order terms, showing that they are nonnegative provided that the norm of the second fundamental form is sufficiently small.

As a consequence, the case in which the norm of the second fundamental form is constantly equal to the threshold value can be completely analyzed, the key being that all of the matrix inequalities used in controlling the zeroth-order terms become equalities. As such, in this setting, the second fundamental form is uniquely determined. As submanifolds of space forms are locally characterized by their first and second fundamental forms, this results in a complete characterization of minimal submanifolds of the round sphere whose second fundamental form is constant and equal to the threshold value. Chern, do Carmo, and Kobayashi's result was later improved by An-Min Li and Jimin Li, making use of the same methods.[5]

In 1973, Kobayashi and Takushiro Ochiai proved some rigidity theorems for Kähler manifolds. In particular, if M is a closed Kähler manifold and there exists α in H1, 1(M, ℤ) such that

${\displaystyle c_{1}(M)\geq (n+1)\alpha ,}$

then M must be biholomorphic to complex projective space. This forms the final part of Yum-Tong Siu and Shing-Tung Yau's proof of the Frankel conjecture.[6] Kobayashi and Ochiai also characterized the situation of c1(M) = nα as M being biholomorphic to a quadratic hypersurface of complex projective space.

## Major publications

Articles

• S.S. Chern, M. do Carmo, and S. Kobayashi. Minimal submanifolds of a sphere with second fundamental form of constant length. Functional Analysis and Related Fields (1970), 59–75. Proceedings of a Conference in honor of Professor Marshall Stone, held at the University of Chicago, May 1968. Springer, New York. Edited by Felix E. Browder. doi:10.1007/978-3-642-48272-4_2
• Shoshichi Kobayashi and Takushiro Ochiai. Characterizations of complex projective spaces and hyperquadrics. J. Math. Kyoto Univ. 13 (1973), 31–47. doi:10.1215/kjm/1250523432

Books

• Foundations of differential geometry (1963, 1969), coauthor with Katsumi Nomizu, Interscience Publishers.
• Reprinted in 1996, from John Wiley & Sons, Inc.
• Hyperbolic Manifolds And Holomorphic Mappings: An Introduction (1970/2005)，World Scientific Publishing Company[7]
• Transformation Groups in Differential Geometry (1972), Springer-Verlag, ISBN 0-387-05848-6
• 曲線と曲面の微分幾何 (1982), 裳華房
• Complex Differential Geometry (1983), Birkhauser
• Differential Geometry of Complex Vector Bundles (1987), Princeton University Press[8]
• 接続の微分幾何とゲージ理論 (1989), 裳華房
• ユークリッド幾何から現代幾何へ (1990), 日本評論社
• Hyperbolic Complex Space (1998)，Springer
• 複素幾何 (2005), 岩波書店
• Differential Geometry of Curves and Surfaces (2019) Springer

## Notes

1. ^ ＵＣバークリー校名誉教授・小林昭七さん死去 (in Japanese). Asahi Shimbun. 2012-09-06. Retrieved 2012-09-16.
2. ^ Jensen, Gary R (2014). "Remembering Shoshichi Kobayashi". Notices of the American Mathematical Society. 61 (11): 1322–1332. doi:10.1090/noti1184.
3. ^ S. Kobayashi (1957). "Theory of Connections". Annali di Matematica Pura ed Applicata. 43: 119–194. doi:10.1007/bf02411907. S2CID 120972987.
4. ^ James Simons. Minimal varieties in Riemannian manifolds. Ann. of Math. (2) 88 (1968), 62–105.
5. ^ Li An-Min and Li Jimin. An intrinsic rigidity theorem for minimal submanifolds in a sphere. Arch. Math. (Basel) 58 (1992), no. 6, 582–594.
6. ^ Yum Tong Siu and Shing Tung Yau. Compact Kähler manifolds of positive bisectional curvature. Invent. Math. 59 (1980), no. 2, 189–204.
7. ^ Griffiths, P. (1972). "Review: Hyperbolic manifolds and holomorphic mappings, by S. Kobayashi". Bull. Amer. Math. Soc. 78 (4): 487–490. doi:10.1090/s0002-9904-1972-12966-5.
8. ^ Okonek, Christian (1988). "Review: Differential geometry of complex vector bundles, by S. Kobayashi". Bull. Amer. Math. Soc. (N.S.). 19 (2): 528–530. doi:10.1090/s0273-0979-1988-15731-x.