Celebratio Mathematica

I have known Shoshi­chi Kobay­ashi since the 1960′s, when I star­ted here as a gradu­ate stu­dent. I have been a fac­ulty mem­ber since 1969, and it is partly thanks to Sho that I am still here. He was chair when I had an of­fer from Cal­tech in the late 1970′s. He very ef­fect­ively con­vinced me to for­sake sunny South­ern Cali­for­nia and re­turn to Berke­ley, on at­tract­ive terms which he ne­go­ti­ated on my be­half. Part of the ar­range­ment was for me to serve as his Vice-Chair for Fac­ulty Ap­point­ments for a year upon my re­turn. This may not sound like much of a prize to many of you who have done that kind of ad­min­is­trat­ive job re­cently. But, in fact, Sho did him­self much of the work him­self which oth­er chairs del­eg­ated to their vice-chairs, so I was very lucky.

I’m very glad that things worked out as they did; among oth­er things, it gave Margo and me many op­por­tun­it­ies to en­joy the com­pany of Sho and his wife, whom we al­ways knew by her very ap­pro­pri­ate Eng­lish name of Grace.

Sho has left a most im­press­ive math­em­at­ic­al leg­acy in the form of a roster of 35 Ph.D. stu­dents, a long list of con­tri­bu­tions to dif­fer­en­tial geo­metry, and many in­flu­en­tial mono­graphs.

Per­haps the most well-known math­em­at­ic­al ob­ject bear­ing his name is the “Kobay­ashi pseudo­met­ric,” which he in­tro­duced in 1967. Des­pite a name which makes it sound like something fake, this is a real meas­ure of dis­tance which quickly be­came in Sho’s hands, and re­mains throughout the math­em­at­ic­al world, an es­sen­tial tool for the study of map­pings between and with­in com­plex man­i­folds.

These are spaces, some of whose dir­ec­tions are para­met­er­ized by “ima­gin­ary num­bers”, but that is not where the “pseudo” comes from. The “pseudo” refers to the fact that, in some spaces, two dif­fer­ent points could have zero dis­tance between them. Sho iden­ti­fied the ab­sence of this un­desir­able prop­erty as one which char­ac­ter­ized cer­tain “good” spaces which he called “hy­per­bol­ic” and which are known as “Kobay­ashi hy­per­bol­ic.”

Sho’s work re­mained con­cen­trated in the area of com­plex geo­metry, where he made a string of fun­da­ment­al con­tri­bu­tions throughout a ca­reer of over fifty years, but he worked in oth­er areas of dif­fer­en­tial geo­metry as well. One of my own pa­pers was a vari­ation on a theme he cre­ated in a pa­per on pos­it­ively curved man­i­folds.

Sho was a mas­ter of math­em­at­ic­al com­mu­nic­a­tion. He even wrote a pa­per called “How to write a math­em­at­ic­al pa­per (in Eng­lish).” (It was writ­ten in Ja­pan­ese.) More im­port­ant, his books, es­pe­cially the two-volume “Found­a­tions of Dif­fer­en­tial Geo­metry” with Kat­sumi Nom­izu, have taught dif­fer­en­tial geo­metry and com­plex geo­metry to gen­er­a­tions of stu­dents and oth­er re­search­ers.

Sho was my per­son­al agent for “open­ing Ja­pan to the West.” Through his col­lab­or­at­or Tak­ushiro Ochi­ai, I was in­vited to vis­it the Uni­versity of Tokyo in the Spring of 1987, and Ja­pan has be­come for me and Margo one of our two fa­vor­ite des­tin­a­tions (along with France, where Sho him­self made his first for­eign math­em­at­ic­al vis­it). We have gone back many many times and even, a couple of times, be­nefited from the col­lec­tion of equip­ment which he and Grace ac­cu­mu­lated for the guest apart­ments of Keio Uni­versity.

We share the grief of the Kobay­ashi fam­ily, es­pe­cially Grace, Mei, and Sumi, whom we have long known, as well as oth­er mem­bers whom we met just today. We are glad that Sho’s passing was a peace­ful one of the kind we all hope for, after a long and ful­filling life, but we will also miss very much his gen­er­ous friend­ship, his sense of hu­mor, and the won­der­ful smile to which Hisashi re­ferred earli­er this morn­ing. For­tu­nately, Sho lives on in the form of his mag­ni­fi­cent math­em­at­ic­al leg­acy and our memor­ies of a won­der­ful man.