Matsushima was appointed as an assistant in the Mathematical Institute of Nagoya Imperial University (like Osaka and other Japanese universities it would soon drop the name "Imperial" from its title) immediately he had received his B.Sc. degree, and he was in post in time for the 1942-43 academic year. There were major difficulties in carrying out research in these war years since, quite apart from military reasons and problems caused by bombing, international mathematical journals were not reaching Japan. Equally, it was very difficult for Japanese mathematicians to publish the results that they were discovering.
The first paper which Matsushima published contained a proof that a conjecture of Zassenhaus was false. Zassenhaus had conjectured that every semisimple Lie algebra L over a field of prime characteristic, with [L, L] = L, is the direct sum of simple ideal and Matsushima was able to construct a counterexample. He then embarked on research which enabled him to prove that Cartan subalgebras of a Lie algebra are conjugate, but due to being out of touch with current research, he was to publish this result while unaware that Chevalley had already published a proof. When he was able to obtain details of another of Chevalley's papers through a review in Mathematical Reviews he was able to construct the proofs for himself.
The 1947 volume of the Proceedings of the Japan Academy, containing two papers by Matsushima, did not appear until 1950 while the first volume of Journal of the Mathematical Society of Japan contained three of his papers. This shows the quantity of mathematics he had managed to produce in the final years of the war and the next couple of years which were almost as difficult ones in which to carry out research in Japan. In 1952-53 he organised a seminar on Lie pseudogroups and differential systems at Nagoya, a topic on which he was now working. One of the students who attended this seminar, Kuranishi, went on to prove a famous result on this topic. Reminiscing on these years later in his life, Matsushima wrote that:-
... although he himself got relatively little out of differential systems for the time and effort spent, he felt much gratified with Kuranishi's great success.Having earlier being promoted to associate professor, in 1953 Matsushima became a full professor at Nagoya University. Chevalley visited him in Nagoya in the autumn of 1953, spending three months there. It was a visit which Matsushima greatly enjoyed, and which was most useful to him. Equally Chevalley enjoyed the visit and invited Matsushima to spend the following year in France. He left for France in the autumn of 1954, spending time at the University of Strasbourg, and then in Paris as a member of C.N.R.S. at the invitation of Chevalley and Henri Cartan. Matsushima presented some of his results to Ehresmann's seminar in Strasbourg, extending Cartan's classification of complex irreducible Lie algebras to the case of real Lie algebras. Having arrived in Paris by the spring of 1955, he lectured on Lie pseudogroups at the Bourbaki seminar. Kobayashi ( or ) writes:-
Matsushima returned to Nagoya in December 1955. His sojourn in France seems to have determined the course of his research for the next several years.When Shoda retired from the Chair of Algebra at Osaka University, Matsushima was appointed to fill the vacant chair in early 1960. His research in Osaka took a somewhat different direction and he wrote a series of papers on cohomology of locally symmetric spaces. In particular he collaborated with Murakami on this topic. In September 1962 he went to the Institute for Advanced Study in Princeton where he spent a year. Back in Osaka, Matsushima jointly began to organise the United States-Japan Seminar in Differential Geometry which was held in Kyoto in June 1965. Shortly after the end of this seminar, he set off for another visit to France to spend the academic year 1965-66 as visiting professor at the University of Grenoble.
In September 1966 Matsushima accepted a chair at the University of Notre Dame in Indiana in the United States. While there he did not end the collaboration which he had been carrying out with Murakami and they continued to undertake joint research. He became an editor of the new Journal of Differential Geometry in 1967, remaining on the editorial board for the rest of his life. Matsushima spent 14 years as a professor at Notre Dame before returning to Japan in 1980. A conference was organised in his honour in May 1980 before he left Notre Dame.
He reached the age of 60 in February 1981 and a volume of papers by colleagues and former students was published: Manifolds and Lie groups, Papers in honour of Yozo Matsushima. The volume contains some papers presented to the conference held in Notre Dame in the previous May. Sadly, after his return to Osaka he had less than three years in his chair before he died from pneumonia at the young age of 62.
Among the honours that Matsushima received, perhaps the most prestigious was the Asahi Prize which was presented to him for his research on continuous groups in 1962.
We have mentioned several topics on which Matsushima worked in the course of this biography. For completeness we list the topics on which Matsushima's contribution is described by in detail by Kobayashi in :-
(1) Lie theory - Lie algebras and Lie groups; Lie pseudogroups and differential systems;
(2) homogeneous complex manifolds - homogeneous Kahler manifolds; homogeneous Stein manifolds; Schubert varieties;
(3) vector bundle valued harmonic forms - Betti numbers of locally symmetric spaces; cohomology of vector bundles over locally symmetric spaces; second fundamental forms and curvature forms;
(4) automorphisms of complex manifolds - automorphisms of Siegel domains; first Chern class and automorphisms of compact Kahler manifolds;
(5) complex tori - vector bundles over complex tori; ample bundles and subvarieties of complex tori; affine structures.
Murakami writes this tribute to Matsushima in  (or see ):-
Matsushima's love for mathematics is evidenced by the discipline and diligence he consistently applied to his research. He was a man whose concern lay not only in mathematics, but he had a keen interest in human nature and a constant curiosity about the world that resulted in his being an avid reader of a variety of books and a most interesting conversationalist. His friends and colleagues will miss not only his mathematical talent, but the warmth and cynical humour which lay behind his outwardly serious countenance.
Article by: J J O'Connor and E F Robertson