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Katsumi Nomizu
Professor of Mathematics (Emeritis)
- CONTACT INFO
Office: 209 Kassar-Gould House
Phone: (401) 863-1133
Fax: (401) 863-9013
Email: nomizu<at>math.brown.edu
Mailing Address:
Mathematics Department
Box 1917
Brown University
Providence, RI 02912
- COURSE SCHEDULE
Retired
- RESEARCH INTERESTS
Differential Geometry
- BACKGROUND
Education: Ph.D., Chicago, 1953; Sc.D., Nagoya (Japan), 1955
- RECENT PUBLICATIONS
Globally defined linear connections on the real line and the circle. Tohoku Math. J. (2) 51 (1999), no. 2, 205--212. w/Sasaki, Takeshi
Selected papers on harmonic analysis, groups, and invariants. Translations from S\=ugaku. Translation edited by Katsumi Nomizu. American Mathematical Society Translations, Series 2, 183. American Mathematical Society, Providence, RI, 1998. x+143 pp. ISBN: 0-8218-0840-0
An introduction to algebraic geometry. Translated from the 1995 Japanese original by Katsumi Nomizu. Translations of Mathematical Monographs, 166. American Mathematical Society, Providence, RI, 1997. xii+246 pp. ISBN: 0-8218-0589-4 by Ueno, Kenji
{\cyr Gruppy Li i differentsial\cprime naya geometriya}. (Russian) [Lie groups and differential geometry] Translated from the English by Yu. A. Shub-Sizonenko. ``PLATON'', Volgograd, 1996. 128 pp. ISBN: 5-80100-130-1
The influence of Katsumi Nomizu on affine differential geometry. Geometry and topology of submanifolds, VII (Leuven, 1994/Brussels, 1994), 33--51, World Sci. Publishing, River Edge, NJ, 1995. by Simon, Udo
Affine connections and their use. Geometry and topology of submanifolds, VII (Leuven, 1994/Brussels, 1994), 197--203, World Sci. Publishing, River Edge, NJ, 1995.
Reflections on some differential geometric work of Katsumi Nomizu. Geometry and topology of submanifolds, VII (Leuven, 1994/Brussels, 1994), 12--32, World Sci. Publishing, River Edge, NJ, 1995.by Dombrowski, Peter
Geometry and topology of submanifolds. VII. Differential geometry in honour of Prof. Katsumi Nomizu. Proceedings of the International Meetings on Pure and Applied Differential Geometry and on Theory of Submanifolds held in Leuven, July 9, 10, 11, 14, and in Brussels, July 12--13, 1994. Edited by F. Dillen, M. Magid, U. Simon, I. Van de Woestijne [I. Van de Woestyne] and L. Verstraelen. World Scientific Publishing Co., Inc., River Edge, NJ, 1995. xii+320 pp. ISBN: 981-02-2195-9
Geometry of surfaces in codimension two. Geometry and topology of submanifolds, V (Leuven/Brussels, 1992), 215--224, World Sci. Publishing, River Edge, NJ, 1993.
Locally symmetric connections on possibly degenerate affine hypersurfaces. Bull. Polish Acad. Sci. Math. 40 (1992), no. 2, 143--150. w/Opozda, Barbara
Foundations of differential geometry. Vol. II. Reprint of the 1969 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1996. xvi+468 pp. ISBN: 0-471-15732-5 w/Kobayashi, Shoshichi
Foundations of differential geometry. Vol. I. Reprint of the 1963 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1996. xii+329 pp. ISBN: 0-471-15733-3. w/Kobayashi, Shoshichi
A survey of recent results in affine differential geometry. Geometry and topology of submanifolds, III (Leeds, 1990), 227--256, World Sci. Publishing, River Edge, NJ, 1991.
Recent results in affine differential geometry---an introduction and a survey. Geometry and global analysis (Sendai, 1993), 351--357, Tohoku Univ., Sendai, 1993.
Selected papers on number theory and algebraic geometry. Translated from S\=ugaku. Translation edited by Katsumi Nomizu. American Mathematical Society Translations, Series 2, 172. American Mathematical Society, Providence, RI, 1996. x+91 pp. ISBN: 0-8218-0445-6
Geodesics in affine differential geometry. Internat. J. Math. 6 (1995), no. 5, 749--766. w/Vrancken, Luc
Affine differential geometry. Geometry of affine immersions. Cambridge Tracts in Mathematics, 111. Cambridge University Press, Cambridge, 1994. xiv+263 pp. ISBN: 0-521-44177-3 w/Sasaki, Takeshi
Recent developments in affine differential geometry. (Japanese) S\=ugaku 46 (1994), no. 4, 308--322.
Another rigidity theorem for affine immersions. Festschrift dedicated to Katsumi Nomizu on his 70th birthday (Leuven, 1994; Brussels, 1994). Results Math. 27 (1995), no. 1-2, 93--96. w/Vrancken, Luc
Selected papers on number theory, algebraic geometry, and differential geometry. Translated from S\=ugaku. Translation edited by Katsumi Nomizu. American Mathematical Society Translations, Series 2, 160. American Mathematical Society, Providence, RI, 1994. x+154 pp. ISBN: 0-8218-7511-6
Selected papers on analysis, probability, and statistics. Translated from S\=ugaku. Translation edited by Katsumi Nomizu. American Mathematical Society Translations, Series 2, 161. American Mathematical Society, Providence, RI, 1994. x+151 pp. ISBN: 0-8218-7512-4
Centroaffine immersions of codimension two and projective hypersurface theory. Nagoya Math. J. 132 (1993), 63--90. w/Sasaki, Takeshi
A new equiaffine theory for surfaces in $\bold R\sp 4$. Internat. J. Math. 4 (1993), no. 1, 127--165. w/Vrancken, Luc
Notes on conjugate connections. Geometry and topology of submanifolds, IV (Leuven, 1991), 152--173, World Sci. Publishing, River Edge, NJ, 1992. w/Simon, Udo
On normal and conormal maps for affine hypersurfaces. Tohoku Math. J. (2) 44 (1992), no. 3, 425--431. w/Opozda, Barbara
Integral formulas for affine surfaces and rigidity theorems of Cohn-Vossen type. Geometry and topology of submanifolds, IV (Leuven, 1991), 133--142, World Sci. Publishing, River Edge, NJ, 1992. w/Opozda, Barbara
On affine hypersurfaces with parallel nullity. J. Math. Soc. Japan 44 (1992), no. 4, 693--699. w/Opozda, Barbara
Affine differential geometry. Proceedings of the conference held at the Mathematisches Forschungsinstitut Oberwolfach, Oberwolfach, February 10--16, 1991. Reprint of Results Math. {20} (1991), no. 3-4 [MR 92i:53002]. Edited by Katsumi Nomizu, Ulrich Pinkall and Udo Simon. Technische Universität Berlin, Berlin, 1991. pp. i--x and 585--810. ISBN: 3-7983-1460-8
Immersions of low rank in differential geometry. Differential geometry, 297--302, Pitman Monogr. Surveys Pure Appl. Math., 52, Longman Sci. Tech., Harlow, 1991. w/Pinkall, Ulrich
A generalization of Lelieuvre's formula. Affine differential geometry (Oberwolfach, 1991). Results Math. 20 (1991), no. 3-4, 682--690. w/Li, An Min; Nomizu, Chang Ping
On the classification of projectively homogeneous surfaces. Affine differential geometry (Oberwolfach, 1991). Results Math. 20 (1991), no. 3-4, 698--724. w/Sasaki, Takeshi
On certain quartic forms for affine surfaces. Tohoku Math. J. (2) 44 (1992), no. 1, 25--33. w/Sasaki, Takeshi
A new model of unimodular-affinely homogeneous surfaces. Manuscripta Math. 73 (1991), no. 1, 39--44. w/Sasaki, Takeshi
On the Cartan-Norden theorem for affine Kähler immersions. Nagoya Math. J. 121 (1991), 127--135. w/Podestà, Fabio
On the geometry of affine Kähler immersions. Nagoya Math. J. 120 (1990), 205--222. w/Pinkall, Ulrich; Podestà, Fabio
Lie groups. I, II. Translated from the Japanese by Katsumi Nomizu. Translations of Mathematical Monographs, 85. American Mathematical Society, Providence, RI, 1991. iv+259 pp. ISBN: 0-8218-4544-6 w/Ise, Mikio; Takeuchi, Masaru
Lorentzian geometry for $2\times 2$ real matrices. Linear and Multilinear Algebra 28 (1991), no. 4, 207--212. w/Pinkall, Ulrich
Conjugate connections and Radon's theorem in affine differential geometry. Monatsh. Math. 109 (1990), no. 3, 221--235. w/Dillen, Franki; Vranken, Luc
On the geometry of projective immersions. J. Math. Soc. Japan 41 (1989), no. 4, 607--623. w/Pinkall, Ulrich
On a theorem of Chern and Terng on affine surfaces. Manuscripta Math. 66 (1990), no. 3, 303--307. w/Sasaki, Takeshi
On affine surfaces whose cubic forms are parallel relative to the affine metric. Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), no. 7, 215--218. w/Magid, Martin A.
Affine hypersurfaces with parallel cubic forms. Tohoku Math. J. (2) 42 (1990), no. 1, 101--108. w/Bokan, Neda; Simon, Udo
On affine Kähler structures. Bull. Soc. Math. Belg. Sér. B 41 (1989), no. 3, 275--282. w/Podestà, F.
Cayley surfaces in affine differential geometry. Tohoku Math. J. (2) 41 (1989), no. 4, 589--596. w/Pinkall, Ulrich
Cubic form theorem for affine immersions. Reprinted in {\it Affine Differentialgeometrie} [(Oberwolfach, 1986), 338--362, Tech. Univ. Berlin, Berlin, 1988. Results Math. 13 (1988), no. 3-4, 338--362. w/Pinkall, Ulrich
On a certain class of homogeneous projectively flat manifolds. Tohoku Math. J. (2) 39 (1987), no. 3, 407--427. w/Pinkall, Ulrich
On the geometry of affine immersions. Math. Z. 195 (1987), no. 2, 165--178. w/Pinkall, Ulrich
On completeness in affine differential geometry. Geom. Dedicata 20 (1986), no. 1, 43--49.
Surfaces of revolution with constant mean curvature in Lorentz\mhy Minkowski space. Tohoku Math. J. (2) 36 (1984), no. 3, 427--437. w/Hano, Jun-ich
Trigonometry in Lorentzian geometry. Amer. Math. Monthly 91 (1984), no. 9, 543--549. w/Birman, Graciela S.
The Gauss\mhy Bonnet theorem for $2$-dimensional spacetimes. Michigan Math. J. 31 (1984), no. 1, 77--81. w/Birman, Graciela S.
Remarks on sectional curvature of an indefinite metric. Proc. Amer. Math. Soc. 89 (1983), no. 3, 473--476.
On isoparametric hypersurfaces in the Lorentzian space forms. Japan. J. Math. (N.S.) 7 (1981), no. 1, 217--226.
On the convexity of spacelike hypersurfaces with nonpositive curvature. Geom. Dedicata 13 (1983), no. 4, 347--350. w/Harris, Steven G.
On isometric immersions of the hyperbolic plane into the Lorentz\mhy Minkowski space and the Monge\mhy Ampère equation of a certain type. Math. Ann. 262 (1983), no. 2, 245--253. w/Hano, Jun-ichi
The Lorentz\mhy Poincaré metric on the upper half-space and its extension. Hokkaido Math. J. 11 (1982), no. 3, 253--261.
{\cyr Osnovy differentsial\cprime no\u\i geometrii. Tom {\rm II}}. (Russian) [Foundations of differential geometry. Vol. II] Translated from the English by L. V. Sabinin. ``Nauka'', Moscow, 1981. 415 pp. w/Kobayasi, Sh.
{\cyr Osnovy differentsial\cprime no\u\i geometrii. Tom {\rm I}}. (Russian) [Foundations of differential geometry. Vol. I] Translated from the English by L. V. Sabinin. With an appendix "Methods of nonassociative geometry" by Sabinin. ``Nauka'', Moscow, 1981. 344 pp. w/Kobayasi, Sh.
On flat surfaces in $S\sp{3}\sb{1}$ and $H\sp{3}\sb{1}$. Manifolds and Lie groups (Notre Dame, Ind., 1980), pp. 71--108, Progr. Math., 14, Birkhäuser, Boston, Mass., 1981. w/Dajczer, Marcos
On the boundedness of Ricci curvature of an indefinite metric. Bol. Soc. Brasil. Mat. 11 (1980), no. 1, 25--30. w/Dajczer, Marcos
On sectional curvature of indefinite metrics. II. Math. Ann. 247 (1980), no. 3, 279--282. w/Dajczer, Marcos
Left-invariant Lorentz metrics on Lie groups. Osaka J. Math. 16 (1979), no. 1, 143--150.
Fundamentals of linear algebra. Second edition. Chelsea Publishing Co., New York, 1979. x+325+30 pp. ISBN: 0-8284-0276-0
Isometric immersions of Lorentzian space forms. Math. Ann. 233 (1978), no. 2, 125--136. w/Graves, Larry
Kinematics and differential geometry of submanifolds. Rolling a ball with a prescribed locus of contact. Tôhoku Math. J. (2) 30 (1978), no. 4, 623--637.
On sectional curvature of indefinite metrics. Math. Ann. 232 (1978), no. 3, 267--272. w/Graves, Larry;
Uniqueness of the normal connections and congruence of isometric immersions. Tôhoku Math. J. (2) 28 (1976), no. 4, 613--617.
Élie Cartan's work on isoparametric families of hypersurfaces. Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 1, Stanford Univ., Stanford, Calif., 1973), pp. 191--200. Amer. Math. Soc., Providence, R.I., 1975.
A characterization of the Veronese varieties. Nagoya Math. J. 60 (1976), 181--188.
On the decomposition of generalized curvature tensor fields. Codazzi, Ricci, Bianchi and Weyl revisited. Differential geometry (in honor of Kentaro Yano), pp. 335--345. Kinokuniya, Tokyo, 1972.
Conditions for constancy of the holomorphic sectional curvature. J. Differential Geometry 8 (1973), 335--339.
On circles and spheres in Riemannian geometry. Math. Ann. 210 (1974), 163--170. w/Yano, Kentaro
Eine Kennzeichnung der projektiven Unterräume in dem komplexen projektiven Raum nach ihren Geodätischen. (German) Math. Z. 137 (1974), 147--150.
Isometric immersions of the hyperbolic plane into the hyperbolic space. Math. Ann. 205 (1973), 181--192.
Characteristic roots and vectors of a differentiable family of symmetric matrices. Linear and Multilinear Algebra 1 (1973), no. 2, 159--162.
A characterization of $n$-spheres in space forms and its complex analogue. Math. Ann. 201 (1973), 75--80.
Generalized central spheres and the notion of spheres in Riemannian geometry. Collection of articles dedicated to Shigeo Sasaki on his sixtieth birthday. Tôhoku Math. J. (2) 25 (1973), 129--137.
Some results in E. Cartan's theory of isoparametric families of hypersurfaces. Bull. Amer. Math. Soc. 79 (1973), 1184--1188.
Umbilical submanifolds and Morse functions. Nagoya Math. J. 48 (1972), 197--201. w/Rodríguez, Lucio
On the spaces of generalized curvature tensor fields and second fundamental forms. Osaka J. Math. 8 (1971), 21--28.
The axiom of spheres in Riemannian geometry. J. Differential Geometry 5 1971 487--489. w/Leung, Dominic S.
A formula of Simons' type and hypersurfaces with constant mean curvature. J. Differential Geometry 3 1969 367--377. w/Smyth, Brian
On the Gauss mapping for hypersurfaces of constant mean curvature in the sphere. Comment. Math. Helv. 44 1969 484--490. w/Smyth, Brian
On the rank and curvature of non-singular complex hypersurfaces in a complex projective space. J. Math. Soc. Japan 21 1969 266--269.
Foundations of differential geometry. Vol. II. Interscience Tracts in Pure and Applied Mathematics, No. 15 Vol. II Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney 1969 xv+470 pp. w/Kobayashi, Shoshichi
Fundamentals of linear algebra. McGraw-Hill Book Co., New York-Toronto, Ont.-London 1966 x+325 pp.
Differential geometry of complex hypersurfaces. II. J. Math. Soc. Japan 20 1968 498--521. w/Smyth, Brian
On hypersurfaces satisfying a certain condition on the curvature tensor. Tôhoku Math. J. (2) 20 1968 46--59.
Some results related to the equivalence problem in Riemannian geometry. Math. Z. 97 1967 29--37. w/Yano, Kentaro
Some results related to the equivalence problem in Riemannian geometry. 1966 Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965) pp. 95--100 Nippon Hyoronsha, Tokyo w/Yano, Kentaro
On infinitesimal transformations preserving the curvature tensor field and its covariant differentials. Ann. Inst. Fourier (Grenoble) 14 1964 fasc. 2, 227--236. w/Yano, Kentaro
Une démonstration simple d'un théorème sur le groupe d'holonomie affine d'un espace de Riemann. (French) C. R. Acad. Sci. Paris 258 1964 5334--5335. w/Yano, Kentaro
Foundations of differential geometry. Vol I. Interscience Publishers, a division of John Wiley & Sons, New York-Lond on 1963 xi+329 pp. w/Kobayashi, Shoshichi
Recent development in the theory of connections and holonomy groups. Advances in Math. 1 1961 fasc. 1, 1--49.
Sur les algèbres de Lie de générateurs de Killing et l'homogénéité d'une variété riemannienne. (French) Osaka Math. J. 14 1962 45--51.
The existence of complete Riemannian metrics. Proc. Amer. Math. Soc. 12 1961 889--891. w/Ozeki, Hideki
On the degree of differentiability of curves used in the definition of the holonomy group. Bull. Amer. Math. Soc. 68 1962 74--75. w/Ozeki, Hideki
A theorem on curvature tensor fields. Proc. Nat. Acad. Sci. U.S.A. 48 1962 206--207. w/Ozeki, Hideki
Holonomy, Ricci tensor and Killing vector fields. Proc. Amer. Math. Soc. 12 1961 594--597.
On local and global existence of Killing vector fields. Ann. of Math. (2) 72 1960 105--120.
{\cyr Gruppy Li i differentsial\cprime naya geometriya}. (Russian) [Lie groups and differential geometry] Translated from the English by Yu. A. \v Sub-Sizonenko; edited by G. F. Laptev. Izdat. Inostr. Lit., Moscow 1960 128 pp.
On Frenet equations for curves of class $C\sp{\infty }$. Tôhoku Math. J. (2) 11 1959 106--112.
On automorphisms of a Kählerian structure. Nagoya Math. J. 11 1957 115--124. w/Kobayashi, Shoshichi;
Remarques sur les groupes d'holonomie et d'isotropie. (French) Colloque de topologie de Strasbourg, 1954--1955, 4 pp. Institut de Mathématique, Université de Strasbourg. no year given.
On infinitesimal holonomy and isotropy groups. Nagoya Math. J. 11 (1957), 111--114.
Lie groups and differential geometry. The Mathematical Society of Japan, 1956. xiv+80 pp.
Un théorème sur les groupes d'holonomie. (French) Nagoya Math. J. 10 (1956), 101--103.
Studies on Riemannian homogeneous spaces. Nagoya Math. J. 9 (1955), 43--56.
Reduction theorem for connections and its application to the problem of isotropy and holonomy groups of a Riemannian manifold. Nagoya Math. J. 9 (1955), 57--66.
On the cohomology of compact homogeneous spaces of nilpotent Lie groups. Ann. of Math. (2) 59, (1954). 531--538.
Sur l'algèbre d'holonomie d'un espace homogène riemannien. (French) C. R. Acad. Sci. Paris 238, (1954). 319--321.
Invariant affine connections on homogeneous spaces. Amer. J. Math. 76, (1954). 33--65.
On the group of affine transformations of an affinely connected manifold. Proc. Amer. Math. Soc. 4, (1953). 816--823.
Application de l'étude des transformations affines aux espaces homogènes riemanniens. (French) C. R. Acad. Sci. Paris 237, (1953). 1386--1387.
Sur les transformations affines d'une variété riemannienne. (French) C. R. Acad. Sci. Paris 237, (1953). 1308--1310.
On the group of automorphisms of a topological group. Tôhoku Math. J. (2) 2, (1950). 47--50. w/Gotô, Morikuni
Quelques résultats en géométrie différentielle des espaces homogènes. (French) [Some results in the differential geometry of homogeneous spaces] Séminaire Bourbaki, Vol. 2, Exp. No. 98, 433--440, Soc. Math. France, Paris, 1995.
These meetings, PADGE and Katsumi Nomizu. Geometry and topology of submanifolds, VII (Leuven, 1994/Brussels, 1994), 52--55, World Sci. Publishing, River Edge, NJ, 1995. by Verstraelen, Leopold
List of publications (August, 1994). Geometry and topology of submanifolds, VII (Leuven, 1994/Brussels, 1994), 7--11, World Sci. Publishing, River Edge, NJ, 1995.
Opportunities and indebtedness. Geometry and topology of submanifolds, VII (Leuven, 1994/Brussels, 1994), 3--6, World Sci. Publishing, River Edge, NJ, 1995.
An introduction to recent results in affine differential geometry [translation of S\=ugaku {46} (1994), no. 4, 308--322; MR 96b:53019]. Sugaku Expositions. Sugaku Expositions 9 (1996), no. 2, 207--225.
{Festschrift dedicated to Katsumi Nomizu on his 70th birthday\}. Papers from the Conference on Differential Geometry held at the Katholieke Universiteit Leuven, Leuven, and the Katholieke Universiteit Brussel, Brussels, July 9--14, 1994. Results Math. {27} (1995), no. 1-2. Birkhäuser Verlag, Basel, 1995. pp. 1--205.
On the classification of projectively homogeneous surfaces. Affine differential geometry (Oberwolfach, 1991), 698--724, Tech. Univ. Berlin, Berlin, 1991. w/Sasaki, Takeshi
A generalization of Lelieuvre's formula. Affine differential geometry (Oberwolfach, 1991), 682--690, Tech. Univ. Berlin, Berlin, 1991. w/Li, An Min; Wang, Chang Ping