Shoshichi Kobayashi
Shoshichi Kobayashi | |
---|---|
Geometry prize (1987) | |
Scientific career | |
Fields | Mathematics |
Institutions | University of California, Berkeley |
Doctoral advisor | Carl B. Allendoerfer |
Doctoral students |
Shoshichi Kobayashi (小林 昭七, Kobayashi Shōshichi, 4 January 1932 – 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
Kobayashi served as chairman of the Berkeley Mathematics Department 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, which he coauthored with Katsumi Nomizu, has been known for its wide influence. In 1970 he was an invited speaker for the section on geometry and topology at the International Congress of Mathematicians in Nice.
Technical contributions
Kobayashi's earliest work dealt with the geometry of connections on principal bundles. Many of these results, along with others, were later absorbed into Foundations of Differential Geometry.
As a consequence of the
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]
On a
Kähler manifold and there exists α in H1, 1(M, ℤ) such thatthen M must be biholomorphic to complex projective space. This, in combination with the Goldberg–Kobayashi result, 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.
Kobayashi is also notable for having proved that a hermitian–Einstein metric on a holomorphic vector bundle over a compact Kähler manifold has deep algebro-geometric implications, as it implies semistability and decomposability as a direct sum of stable bundles.[7] This establishes one direction of the Kobayashi–Hitchin correspondence. Karen Uhlenbeck and Yau proved the converse result, following well-known partial results by Simon Donaldson.
In the 1960s, Kobayashi introduced what is now known as the Kobayashi metric. This associates a pseudo-metric to any complex manifold, in a holomorphically invariant way.[8] This sets up the important notion of Kobayashi hyperbolicity, which is defined by the condition that the Kobayashi metric is a genuine metric (and not only a pseudo-metric). With these notions, Kobayashi was able to establish a higher-dimensional version of the Alhfors–Schwarz lemma from complex analysis.
Major publications
Articles
- Kobayashi, Shoshichi (1959). "Geometry of bounded domains". Zbl 0136.07102.
- Goldberg, Samuel I.; Kobayashi, Shoshichi (1967). "Holomorphic bisectional curvature". Zbl 0169.53202.
- Zbl 0216.44001.
- Kobayashi, Shoshichi; Ochiai, Takushiro (1973). "Characterizations of complex projective spaces and hyperquadrics". Journal of Mathematics of Kyoto University. 13 (1): 31–47. Zbl 0261.32013.
- Kobayashi, Shoshichi (1976). "Intrinsic distances, measures and geometric function theory" (PDF). Zbl 0346.32031.
Books
- Kobayashi, Shoshichi; Zbl 0119.37502.[9]
- Kobayashi, Shoshichi; Zbl 0175.48504.
- Kobayashi, Shoshichi (1972). Transformation groups in differential geometry. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 70. New York–Heidelberg: Zbl 0246.53031.
- Kobayashi, Shoshichi (1984). An introduction to the theory of connections. Seminar on Mathematical Sciences. Vol. 8. Notes by Kotaro Yamada. Yokohama: Zbl 0547.53018.
- Kobayashi, Shoshichi (1987). Differential geometry of complex vector bundles. Publications of the Mathematical Society of Japan. Vol. 15. Reprinted in 2014. Princeton, NJ: Zbl 0708.53002.[10]
- Kobayashi, Shoshichi (1998). Hyperbolic complex spaces. Grundlehren der mathematischen Wissenschaften. Vol. 318. Berlin: Zbl 0917.32019.
- Kobayashi, Shoshichi (2005). Hyperbolic manifolds and holomorphic mappings. An introduction (Second edition of 1970 original ed.). Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd. Zbl 1084.32018.[11]
- Kobayashi, Shoshichi (2019). Differential geometry of curves and surfaces. Springer Undergraduate Mathematics Series. Translated by Nagmo, Eriko Shinozaki; Tanaka, Makiki Sumi (Revised edition of 1977 original ed.). Singapore: Zbl 1437.53001.
Kobayashi was also the author of several textbooks which (as of 2022) have only been published in Japanese.[12]
Notes
- Asahi Shimbun. 2012-09-06. Retrieved 2012-09-16.
- doi:10.1090/noti1184.
- S2CID 120972987.
- ^ James Simons. Minimal varieties in Riemannian manifolds. Ann. of Math. (2) 88 (1968), 62–105.
- ^ 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.
- ^ Yum Tong Siu and Shing Tung Yau. Compact Kähler manifolds of positive bisectional curvature. Invent. Math. 59 (1980), no. 2, 189–204.
- ^ Kobayashi 1987, Theorem 5.8.3.
- ^ Kobayashi 2005.
- MR 1566282.
- .
- .
- ^ Books authored by Shoshichi Kobayashi
References
- 特集◎小林昭七 [Feature: Kobayashi Shoshichi]. 数学セミナー (in Japanese). February 2013. Archived from the original on 2013-01-26.
External links
- Shoshichi Kobayashi – In Memoriam
- Publications of Shoshichi Kobayashi
- Shoshichi Kobayashi Department of Mathematics UC Berkeley
- Shoshichi Kobayashi at the Mathematics Genealogy Project