was a Japanese

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

. He was the eldest brother of electrical engineer and computer scientist Hisashi Kobayashi. His research interests were in Riemannian and

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

s, transformation groups of geometric structures, and

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

Biography

Kobayashi graduated from the

University of Tokyo The University of Tokyo (, abbreviated as in Japanese and UTokyo in English) is a public research university in Bunkyō, Tokyo, Japan. Founded in 1877 as the nation's first modern university by the merger of several pre-westernisation era ins ...

in 1953. In 1956, he earned a Ph.D. from the

University of Washington The University of Washington (UW and informally U-Dub or U Dub) is a public research university in Seattle, Washington, United States. Founded in 1861, the University of Washington is one of the oldest universities on the West Coast of the Uni ...

under Carl B. Allendoerfer. His dissertation was ''Theory of Connections''. He then spent two years at the

Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...

and two years at

MIT The Massachusetts Institute of Technology (MIT) is a private research university in Cambridge, Massachusetts, United States. Established in 1861, MIT has played a significant role in the development of many areas of modern technology and sc ...

. He joined the faculty of the

University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California), is a Public university, public Land-grant university, land-grant research university in Berkeley, California, United States. Founded in 1868 and named after t ...

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 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 ''Foundations of Differential Geometry'' is an influential 2-volume mathematics book on differential geometry written by Shoshichi Kobayashi and Katsumi Nomizu. The first volume was published in 1963 and the second in 1969, by Interscience Publis ...

'', which he coauthored with

Katsumi Nomizu was a Japanese-American mathematician known for his work in differential geometry. Life and career Nomizu was born in Osaka, Japan on December 1, 1924. He studied mathematics at Osaka University, graduating in 1947 with a Master of Science th ...

, 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 The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before ...

Nice Nice ( ; ) is a city in and the prefecture of the Alpes-Maritimes department in France. The Nice agglomeration extends far beyond the administrative city limits, with a population of nearly one millionconnections on principal bundles. Many of these results, along with others, were later absorbed into ''

''. As a consequence of the

Gauss–Codazzi equations In Riemannian geometry and pseudo-Riemannian geometry, the Gauss–Codazzi equations (also called the Gauss–Codazzi–Weingarten-Mainardi equations or Gauss–Peterson–Codazzi formulas) are fundamental formulas that link together the induced m ...

and the commutation formulas for

covariant derivative In mathematics and physics, covariance is a measure of how much two variables change together, and may refer to: Statistics * Covariance matrix, a matrix of covariances between a number of variables * Covariance or cross-covariance between ...

s, James Simons discovered a formula for the Laplacian of the

second fundamental form In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by \mathrm (read "two"). Together with the first fundamen ...

of a submanifold of a

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

. 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 In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The ...

of the submanifold is zero and when the Riemannian manifold has constant curvature. In this setting,

Shiing-Shen Chern Shiing-Shen Chern (; , ; October 26, 1911 – December 3, 2004) was a Chinese American mathematician and poet. He made fundamental contributions to differential geometry and topology. He has been called the "father of modern differential geome ...

Manfredo do Carmo Manfredo Perdigão do Carmo (15 August 1928, Maceió – 30 April 2018, Rio de Janeiro) was a Brazilian mathematician. He spent most of his career at IMPA and is seen as the doyen of differential geometry in Brazil. Education and career Do Ca ...

, 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 form In mathematics, a space form is a complete Riemannian manifold ''M'' of constant sectional curvature ''K''. The three most fundamental examples are Euclidean ''n''-space, the ''n''-dimensional sphere, and hyperbolic space, although a space form n ...

s 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. On a

Kähler manifold In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnol ...

, it is natural to consider the restriction of the

sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a po ...

to the two-dimensional planes which are holomorphic, i.e. which are invariant under the almost-complex structure. This is called the ''holomorphic sectional curvature''. Samuel Goldberg and Kobayashi introduced an extension of this quantity, called the ''holomorphic bisectional curvature''; its input is a pair of holomorphic two-dimensional planes. Goldberg and Kobayashi established the differential-geometric foundations of this object, carrying out many analogies with the sectional curvature. In particular they established, by the Bochner technique, that the second

Betti number In algebraic topology, the Betti numbers are used to distinguish topological spaces based on the connectivity of ''n''-dimensional simplicial complexes. For the most reasonable finite-dimensional spaces (such as compact manifolds, finite simplicia ...

of a connected

closed manifold In mathematics, a closed manifold is a manifold Manifold with boundary, without boundary that is Compact space, compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The onl ...

must equal one if there is a Kähler metric whose holomorphic bisectional curvature is positive. Later, Kobayashi and Takushiro Ochiai proved some rigidity theorems for

s. In particular, if is a closed Kähler manifold and there exists in such that :

c_1(M)\geq(n+1)\alpha,

then must be biholomorphic to

complex projective space In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...

. This, in combination with the Goldberg–Kobayashi result, forms the final part of

Yum-Tong Siu Yum-Tong Siu (; born May 6, 1943) is a Chinese mathematician. He is the William Elwood Byerly Professor of Mathematics at Harvard University. Siu is a prominent figure in the study of functions of several complex variables. His research interes ...

and

Shing-Tung Yau Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician. He is the director of the Yau Mathematical Sciences Center at Tsinghua University and professor emeritus at Harvard University. Until 2022, Yau was the William Caspar ...

's proof of the Frankel conjecture. Kobayashi and Ochiai also characterized the situation of as 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 In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a ...

over a compact Kähler manifold has deep algebro-geometric implications, as it implies semistability and decomposability as a direct sum of stable bundles. This establishes one direction of the

Kobayashi–Hitchin correspondence In differential geometry, algebraic geometry, and gauge theory, the Kobayashi–Hitchin correspondence (or Donaldson–Uhlenbeck–Yau theorem) relates stable vector bundles over a complex manifold to Einstein–Hermitian vector bundles. The corres ...

Karen Uhlenbeck Karen Keskulla Uhlenbeck ForMemRS (born August 24, 1942) is an American mathematician and one of the founders of modern geometric analysis. She is a professor emeritus of mathematics at the University of Texas at Austin, where she held the Sid W ...

and Yau proved the converse result, following well-known partial results by

Simon Donaldson Sir Simon Kirwan Donaldson (born 20 August 1957) is an English mathematician known for his work on the topology of smooth function, smooth (differentiable) four-dimensional manifolds, Donaldson–Thomas theory, and his contributions to Kähl ...

. In the 1960s, Kobayashi introduced what is now known as the

Kobayashi metric In mathematics and especially complex geometry, the Kobayashi metric is a pseudometric intrinsically associated to any complex manifold. It was introduced by Shoshichi Kobayashi in 1967. Kobayashi hyperbolic manifolds are an important class of com ...

. This associates a pseudo-metric to any

complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ...

, in a holomorphically invariant way. 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 Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...

Major publications

Articles * * * * * Books * * * * * * * * Kobayashi was also the author of several textbooks which (as of 2022) have only been published in Japanese.Books authored by Shoshichi Kobayashi
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Notes

References

External links

Publications of Shoshichi Kobayashi

Shoshichi Kobayashi
Department of Mathematics UC Berkeley * {{DEFAULTSORT:Kobayashi, Shoshichi Differential geometers University of Tokyo alumni University of Washington alumni University of California, Berkeley College of Letters and Science faculty 1932 births 2012 deaths 20th-century Japanese mathematicians 21st-century Japanese mathematicians People from Kōfu, Yamanashi Scientists from Yamanashi Prefecture Japanese emigrants to the United States