# Katsumi Nomizu

**NomizuNomizu, Katsumi**

Katsumi Nomizu was a Japanese-American mathematician known for his work in differential geometry.wikipedia

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### Foundations of Differential Geometry

During this time he embarked on a major collaborative project with Professor Shoshichi Kobayashi at the University of California, Berkeley, resulting in the classic two-volume work, Foundations of Differential Geometry in 1963.

Foundations of Differential Geometry is an influential 2-volume mathematics book on differential geometry written by Shoshichi Kobayashi and Katsumi Nomizu.

### Shiing-Shen Chern

**S. S. ChernChernChen Xingshen**

He studied first at Columbia University and then at the University of Chicago where in 1953 he became the first student to earn a Ph.D. under the thesis direction of Shiing-Shen Chern.

Two of his students Manfredo do Carmo and Katsumi Nomizu have written influential textbooks in geometry.

### Shoshichi Kobayashi

**KobayashiKobayashi, Shoshichi**

During this time he embarked on a major collaborative project with Professor Shoshichi Kobayashi at the University of California, Berkeley, resulting in the classic two-volume work, Foundations of Differential Geometry in 1963.

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

### Kulkarni–Nomizu product

**Kulkarni-Nomizu product**

In the mathematical field of differential geometry, the Kulkarni–Nomizu product (named for Ravindra Shripad Kulkarni and Katsumi Nomizu) is defined for two (0,2)-tensors and gives as a result a (0,4)-tensor.

### Mathematician

**mathematiciansapplied mathematicianMathematics**

Katsumi Nomizu was a Japanese-American mathematician known for his work in differential geometry.

### Differential geometry

**differentialdifferential geometerdifferential geometry and topology**

Katsumi Nomizu was a Japanese-American mathematician known for his work in differential geometry.

### Osaka

**Osaka, JapanŌsakaOsaka, Osaka**

Nomizu was born in Osaka, Japan on the first day of December, 1924.

### Mathematics

**mathematicalmathmathematician**

He studied mathematics at Osaka University, graduating in 1947 with a master of science then traveled to the United States on a U.S. Army Fulbright Scholarship.

### Osaka University

**University of OsakaOsaka Imperial UniversityOsaka**

He studied mathematics at Osaka University, graduating in 1947 with a master of science then traveled to the United States on a U.S. Army Fulbright Scholarship.

### Master of Science

**M.S.MScMS**

He studied mathematics at Osaka University, graduating in 1947 with a master of science then traveled to the United States on a U.S. Army Fulbright Scholarship.

### Nagoya University

**University of NagoyaNagoyaNagoya Imperial University**

Returning to Japan, he studied at Nagoya University, obtaining a doctor of science in 1955.

### Doctor of Science

**DScD.Sc.Sc.D.**

Returning to Japan, he studied at Nagoya University, obtaining a doctor of science in 1955.

### Brown University

**BrownCollege in the English Colony of Rhode Island and Providence PlantationsBrown Association for Cooperative Housing**

In 1960 he began his thirty-five-year career with Brown University, first as associate professor, then becoming full professor in 1963.

### Associate professor

**Associateassociate professorsAssoc. Prof.**

In 1960 he began his thirty-five-year career with Brown University, first as associate professor, then becoming full professor in 1963.

### Doctor of Philosophy

**Ph.D.PhDPh.D**

In his teaching career he helped 13 students from Brown and 1 from MIT obtain a Ph.D. According to the Mathematics Genealogy Project, Professor Nomizu has seventeen mathematical grandchildren.

### American Mathematical Society

**AMSChelsea PublishingNew York Mathematical Society**

He served as editor of a collection of papers on number theory and algebraic geometry published by the American Mathematical Society in 1996.

### Providence, Rhode Island

**ProvidenceProvidence, RIProvidence, R.I.**

Professor Nomizu died on November 5, 2008, in Providence, Rhode Island.

### Humboldt Prize

**Humboldt Research AwardAlexander von Humboldt Research AwardAlexander von Humboldt Award**

### Ancestry.com

**RootsWebNewspapers.comRootsWeb.com**

### Mathematics Genealogy Project

**descendantsmathematical genealogyMathematical Genealogy Project**

### Ernst Ruh

**Ernst Alfred RuhRuh, Ernst**

Ernst Ruh received his doctorate in 1964 from Brown University under Katsumi Nomizu with thesis On the Automorphism Groups of a G-structure.

### Riemannian connection on a surface

**Cartan's first and second structure equationscovariant surface derivativesuccinctly expressed and derived**

Cartan's approach was rephrased in the modern language of principal bundles by Ehresmann, after which the subject rapidly took its current form following contributions by Chern, Ambrose and Singer, Kobayashi, Nomizu, Lichnerowicz and others.

### Affine geometry

**affineaffine ''d''-spaceaffine geometries**

In 1984, "the affine plane associated to the Lorentzian vector space L 2 " was described by Graciela Birman and Katsumi Nomizu in an article entitled "Trigonometry in Lorentzian geometry".

### Symmetric space

**Riemannian symmetric spacesymmetric spaceslocally symmetric space**

As shown by Katsumi Nomizu, there is a dichotomy: an irreducible symmetric space G/H is either flat (i.e., an affine space) or \mathfrak g is semisimple.

### Curvature form

**Bianchi identityBianchi identitiescurvature**

* Shoshichi Kobayashi and Katsumi Nomizu (1963) Foundations of Differential Geometry, Vol.I, Chapter 2.5 Curvature form and structure equation, p 75, Wiley Interscience.