Celebratio Mathematica

The life and aca­dem­ic achieve­ments of Shoshi­chi Kobay­ashi give a defin­i­tion of what a great math­em­atician should be. He left us nu­mer­ous manuscripts of high ori­gin­al­ity, com­par­able to great mu­sic­al com­pos­i­tions. His books, taken as a whole, har­mon­ize splen­didly in­to a great sym­phony. Though aware of my in­ab­il­ity to reach the height of his tal­ent, I dare to write this art­icle to bring at­ten­tion to the fine per­son­al char­ac­ter and re­mark­able aca­dem­ic achieve­ments of Shoshi­chi Kobay­ashi, out­stand­ing math­em­atician, ment­or, and col­league.

Kobay­ashi pub­lished pa­pers in aca­dem­ic journ­als every year without fail, start­ing from his vir­gin pa­per in 1954 un­til his last days. Among a total of 134 pa­pers, there are 85 single-au­thor pa­pers and 49 col­lab­or­at­ive works. A unique fea­ture of his single-au­thor pub­lic­a­tions is their brev­ity. There are 34 pa­pers of less than five pages, 28 pa­pers of less than ten pages, and 15 pa­pers of less than 28 pages. Every one of them is deep and rich in con­tent with trans­par­ent and eas­ily com­pre­hens­ible ex­plan­a­tions, as is evid­enced by his in­ven­tion of the con­cepts of the Kobay­ashi dis­tance, hy­per­bol­ic com­plex man­i­folds, and the Her­mitian–Ein­stein vec­tor bundles. In­spired by Chern’s res­ult, which fur­ther im­proved on the gen­er­al­iz­a­tion of the clas­sic­al Schwarz’s lemma by L. Ahlfors (“The holo­morph­ic maps between Her­mitian man­i­folds” by S. S. Chern), Kobay­ashi be­came in­ter­ested in Schwarz’s lemma and read all the re­lated pa­pers one by one. He es­pe­cially ad­mired Carathéodory’s point of view and de­voted him­self to and took ad­vant­age of the holo­morph­ic maps, which he ad­mit­ted led him to the dis­cov­ery of the Kobay­ashi dis­tance.

Ex­pand­ing all his pa­pers, pay­ing at­ten­tion to the his­tor­ic back­ground and de­vel­op­ment, Kobay­ashi sub­sequently pub­lished thir­teen self-con­tained books, every one of which is eas­ily com­pre­hens­ible by gradu­ate stu­dents. Pro­fess­or Kobay­ashi once said to me: “When I write books, I pre­pare one or two years for the first manuscript and make cer­tain to give lec­tures based on it for one or two years in or­der to deep­en the con­tents be­fore fi­nally com­plet­ing the fi­nal manuscript.”

To con­vey to the pub­lic his philo­sophy of math­em­at­ics, or rather of dif­fer­en­tial geo­metry, I in­clude here pas­sages from his es­says in Ja­pan­ese as pub­lished in well-known magazines on math­em­at­ic­al sci­ences which deeply re­flect his math­em­at­ic­al achieve­ments. The trans­par­ency, beauty, and open-minded­ness of his view­points, which he sharpened well to the lim­it, ex­press my per­son­al and etern­al yearn­ing.

Suuri Kagaku, Au­gust, 1965:
Dif­fer­en­tial geo­metry is one view­point over math­em­at­ics, and a meth­od, in it­self. When vari­ous fields in math­em­at­ics reach the point of be­ing well un­der­stood, the phe­nomen­on called al­geb­ra­iz­a­tion oc­curs. I think it is pos­sible to see dif­fer­ent fields of math­em­at­ics from a dif­fer­en­tial geo­met­ric view­point as well as from an al­geb­ra­ic view­point. The rais­on d’être of dif­fer­en­tial geo­metry is to of­fer a new view­point and power­ful meth­ods rather than be­ing con­sidered like a the­ory of num­bers. Moreover, the con­cepts and meth­ods un­der­stood from a geo­met­ric point of view are so nat­ur­al (dif­fer­ent from ar­ti­fi­cially cre­ated non­sense) that de­vel­op­ment bey­ond an­ti­cip­a­tion later oc­curs in many cases. Dif­fer­en­tial geo­metry can pro­duce lim­it­less de­vel­op­ments by car­ry­ing its meth­ods in­to all the fields of math­em­at­ics. It is es­pe­cially im­port­ant, then, to know how to make use of the dif­fer­en­tial geo­met­ric meth­od to make con­nec­tions in fields such as the the­ory of func­tions (one or sev­er­al com­plex vari­ables), al­geb­ra­ic geo­metry, to­po­logy, a dif­fer­en­tial equa­tion the­ory.

Sūgaku Sem­in­ar, Decem­ber, 1965:
Won­der­ful the­or­ems in math­em­at­ics are proven with break­throughs of ori­gin­al­ity which cause every­one to un­der­stand them. A math­em­atician gets the greatest feel­ing of hap­pi­ness when he finds such a new idea. A prob­lem, which is solved merely by un­der­stand­ing and fol­low­ing a routine pro­cess, is a petty prob­lem, and a the­or­em whose proof does not use any idea which comes in­vol­un­tar­ily, is really te­di­ous. It seems that any prob­lem which does not open a new field, or any res­ult which has no ap­plic­a­tion in any oth­er field of math­em­at­ics, dis­ap­pears after a while. An etern­al life is giv­en only to a beau­ti­ful res­ult.

Sūgaku Sem­in­ar, Septem­ber, 1967:
Some per­cent­age of the work in math­em­at­ics con­sists of the cre­ation of suit­able nota­tion. Suit­able nota­tion makes cal­cu­la­tions easy to handle, makes for­mu­las look beau­ti­ful and easy to mem­or­ize, and makes the­or­ems ap­par­ent at a glance. Suit­able nota­tion is im­port­ant not only from a pass­ive role to make de­scrip­tions beau­ti­ful and easy, but also from an act­ive role, even to sug­gest­ing what should be done next. We can­not ex­plain in one word what kind of nota­tion is suit­able or what kind of the­or­em is good. It is a math­em­at­ic­al sense that will take it in some­how.

Suuri Kagaku, Feb­ru­ary, 2001:
When you study sub­jects in math­em­at­ics which are not re­stric­ted to cal­cu­lus, I would like to urge you to study the his­tory of its de­vel­op­ment to­geth­er by all means. Since mod­ern math­em­at­ics re­quires too much strin­gency, lec­tures as well as text­books rarely touch his­tor­ic­al back­ground. However, I think that one can un­der­stand the sub­ject more deeply by get­ting to know the his­tory and why this concept was pro­duced. I want people who be­come school teach­ers to study the his­tory of math­em­at­ics by all means.

Through my ac­quaint­ance with Pro­fess­or Kobay­ashi for many years, I am con­vinced that his pas­sion for math­em­at­ics was mo­tiv­ated by his love for hu­man be­ings and for schol­ar­ship. I am ex­tremely for­tu­nate to have col­lab­or­ated with Pro­fess­or Kobay­ashi on some of his math­em­at­ic­al work.