Celebratio Mathematica

Shōshichi Kobayashi


Shoshi­chi Kobay­ashi, 80, Emer­it­us Pro­fess­or of Math­em­at­ics at the Uni­versity of Cali­for­nia at Berke­ley, died peace­fully in his sleep on Au­gust 29, 2012. He was on the fac­ulty at Berke­ley for 50 years, and has au­thored over 15 books in the area of dif­fer­en­tial geo­metry and the his­tory of math­em­at­ics.

Shoshi­chi stud­ied at the Uni­versity of Tokyo, re­ceiv­ing his B.S. de­gree in 1953. He spent one year of gradu­ate study in Par­is and Stras­bourg (1953–54), and com­pleted his Ph.D. at the Uni­versity of Wash­ing­ton, Seattle in 1956. He was ap­poin­ted Mem­ber of the In­sti­tute for Ad­vanced Study at Prin­ceton (1956–58), Postdoc­tor­al Re­search As­so­ci­ate at MIT (1958–60), and As­sist­ant Pro­fess­or at the Uni­versity of Brit­ish Columbia (1960–62). In 1962 he joined the fac­ulty at Berke­ley and be­came Full Pro­fess­or in 1966.

He was a vis­it­ing pro­fess­or at nu­mer­ous de­part­ments of math­em­at­ics around the world, in­clud­ing the Uni­versity of Tokyo, the Uni­versity of Mainz, the Uni­versity of Bonn, MIT and the Uni­versity of Mary­land. Most re­cently he had been vis­it­ing Keio Uni­versity in Tokyo. He was a Sloan Fel­low (1964–66), a Gug­gen­heim Fel­low (1977–78) and Chair­man of his De­part­ment (1978–81).

Shoshi­chi Kobay­ashi was one of the greatest con­trib­ut­ors to the field of dif­fer­en­tial geo­metry in the last half of the twen­ti­eth cen­tury.

His early work, be­gin­ning in 1954, con­cerned the the­ory of con­nec­tions, a no­tion ba­sic to all as­pects of dif­fer­en­tial geo­metry and its ap­plic­a­tions. Prof. Kobay­ashi’s early work was es­sen­tially in cla­ri­fy­ing and ex­tend­ing many of Élie Cartan‘s ideas, par­tic­u­larly those in­volving pro­ject­ive and con­form­al geo­metry, and mak­ing them avail­able to mod­ern dif­fer­en­tial geo­met­ers. A second ma­jor in­terest of his was the re­la­tion of curvature to to­po­logy, in par­tic­u­lar for Kähler man­i­folds.

Throughout his ca­reer, Prof. Kobay­ashi con­tin­ued to fo­cus his at­ten­tion on Kähler and more gen­er­al com­plex man­i­folds. One of his most en­dur­ing con­tri­bu­tions was the in­tro­duc­tion in 1967 of what soon be­came known as the “Kobay­ashi pseudodistance,” along with the re­lated no­tion of “Kobay­ashi hy­per­bol­i­city.” Since that time, these no­tions have be­come in­dis­pens­able tools for the study of map­pings of com­plex man­i­folds.

Oth­er areas in which Kobay­ashi made fun­da­ment­al ad­vances in­clude the the­ory of com­plex vec­tor bundles, in­trins­ic dis­tances in af­fine and pro­ject­ive dif­fer­en­tial geo­metry, and the study of the sym­met­ries of geo­met­ric struc­tures us­ing filtered Lie al­geb­ras.

Sev­er­al of Shoshi­chi Kobay­ashi’s books are stand­ard ref­er­ences in dif­fer­en­tial and com­plex geo­metry, among them his two-volume treat­ise with Kat­sumi Nom­izu en­titled “Found­a­tions of Dif­fer­en­tial Geo­metry.” Gen­er­a­tions of stu­dents and oth­er schol­ars have learned the es­sen­tials of the sub­ject from his books.

The fol­low­ing is a trans­la­tion by Prof. Toshiki Mabu­chi (Osaka Uni­versity) of his 1992 de­scrip­tion of Shoshi­chi Kobay­ashi’s work.

His books “Found­a­tions of Dif­fer­en­tial Geo­metry, Volumes I & II” coau­thored by K. Nom­izu are very pop­u­lar not only among math­em­aticians but also among phys­i­cists.

His book on hy­per­bol­i­city and trans­form­a­tion groups also in­flu­enced many math­em­aticians.

He pub­lished more than one hun­dred pa­pers, which have re­ceived an ex­cep­tion­ally large num­ber of cita­tions.

His math­em­at­ic­al achieve­ments range across dif­fer­en­tial geo­metry, Lie al­geb­ras, trans­form­a­tion groups and com­plex ana­lys­is. The most im­port­ant ones are:

  1. Kobay­ashi’s in­trins­ic pseudo-dis­tance and its dis­tance-de­creas­ing prop­erty for holo­morph­ic map­pings;
  2. Kobay­ashi hy­per­bol­i­city;
  3. Meas­ure hy­per­bol­i­city and the gen­er­al­ized Schwarz lemma;
  4. Pro­ject­ively in­vari­ant dis­tances for af­fine and pro­ject­ive dis­tances;
  5. The study of com­pact com­plex man­i­folds with pos­it­ive Ricci curvature and Kobay­ashi–Ochi­ai’s char­ac­ter­iz­a­tion of com­plex pro­ject­ive spaces and hy­per­quad­rics:
  6. Filtered Lie al­geb­ras and geo­met­ric struc­tures;
  7. The study of Her­mitian–Ein­stein holo­morph­ic vec­tor bundles and the Kobay­ashi-Hitchin cor­res­pond­ence.

In (1), (2) and (3), we see his ex­tremely high ori­gin­al­ity, and (5) has led suc­ceed­ing math­em­aticians to Frankel’s con­jec­tures, which (7) have had had great im­pact on al­geb­ra­ic geo­metry as well as dif­fer­en­tial geo­metry — Tian–Don­ald­son–Yau’s Con­jec­ture on the \( K \)-sta­bil­ity and ex­ist­ence of Kähler–Ein­stein met­rics is still a cent­ral prob­lem in com­plex geo­metry.