by Gary R. Jensen
For me, Shoshichi Kobayashi was the ideal thesis advisor and mentor. Without his help and guidance at many stages, I would never have become a professor of mathematics. Here is the story.
After four semesters of course work at Berkeley, I could imagine only Professor Kobayashi as someone I might talk to. When I asked him to be my advisor, his response was cautious but not cold. After all, I had taken no courses in differential geometry other than the required curves and surfaces course. He suggested that I spend the summer reading his new book, written with Nomizu, Foundations of Differential Geometry, vol. I. Thus began my lifelong love of differential geometry.
At the beginning of the fall semester he agreed to be my advisor. He handed me a list of problems he and James Eells had edited for the proceedings of a recent conference in Kyoto, Japan, with an indication of two or three problems that might interest me. A week later we agreed that I look at the problem proposed by Eells and Sampson: Does any simply connected, compact Riemannian space of nonnegative curvature admit a Ricci parallel metric? During that academic year I read papers and got nowhere. When I finally told Professor Kobayashi that I felt I was making no progress, in fact, that I really had no idea of how to begin trying to solve this problem, his reply was simple and profoundly helpful. “Why don’t you just look at the dimension four homogeneous case first,” he said. It is embarrassing to remember that for a year this idea had not occurred to me. Along with this advice he suggested a paper by S. Ishihara on four-dimensional homogeneous spaces. At last, a paper I understood and saw how to apply to my problem.
Graduate school at Berkeley in the mid-sixties was wonderful. In our meeting at the beginning of the fall 1967 semester, Professor Kobayashi quietly mentioned that this would be my last year. “But I don’t have enough results for a thesis,” I reminded him. Yes, he agreed, but he stated again that this would be my last year. I got the message and it was a powerful motivator. In our weekly meetings I started presenting partial results, bits and pieces that at first seemed to hit a wall, but soon began yielding to the assault. By the end of December I had found all homogeneous Einstein spaces of dimension four.
Meanwhile, Professor Kobayashi had raised the issue of a job for next year. Strangely enough, I was very vague on this point, no doubt due to my subconscious desire to remain a graduate student for the rest of my life. He introduced me to some mathematicians from a university in the East, and I told them I would like to join their department. Weeks passed with no communication from that department. I didn’t give it much thought, but one day Professor Kobayashi asked me, with some anxiety, whether I had heard anything. Hearing the answer and hearing that I had not applied for anything else, he took me by the arm and escorted me to the library’s collection of notebooks of available jobs. Together we picked out a few. He told me to write a letter of application to each and to find two more people to write letters of recommendation for me. It still frightens me to think what might have become of me if he had not intervened so effectively at that time to make sure that I had applied for jobs. In early March I interviewed at Carnegie-Mellon and accepted their offer.
In June my family and I headed out for Pittsburgh with a copy of the manuscript of Foundations of Differential Geometry, vol. II, in the trunk of the car. After reading it I struggled to find new research problems. I wrote to Professor Kobayashi that I needed more contact with differential geometers. I asked him if a postdoctoral fellowship somewhere might be possible. In early March a call came from Washington University in St. Louis asking me if I would be interested in coming there to interview for a one-year postdoc position connected to a special year in symmetric spaces. Professor Kobayashi had suggested my name to them for this position.
At the end of my postdoc year, I accepted a tenure-track offer from Washington University. Professor Kobayashi’s mentoring continued. He was instrumental in arranging a visiting research position for me at Berkeley during the summer of 1971. In one conversation that summer he directed my attention to his 1963 Tôhoku Math. J. paper “Topology of positively pinched Kaehler manifolds,” which formed the basis of my best-known paper of the 1970s, “Einstein metrics on principal fibre bundles,” a paper that probably tipped the tenure decision in my favor.
