Manfredo do Carmo

Manfredo do Carmo Manfredo do Carmo was a Brazilian mathematician renowned for his work in differential geometry, especially the theory of minimal surfaces and the geometry of submanifolds. He held influential positions at major institutions and authored foundational texts widely used in courses on Riemannian geometry, curvature theory, and global differential geometry. His work connected classical problems treated by figures such as Carl Friedrich Gauss, Bernhard Riemann, and Henri Poincaré with modern developments associated with Shiing-Shen Chern and Marston Morse.

Early life and education

Born in Rio de Janeiro in 1928, he completed his undergraduate studies during a period when Brazilian universities were expanding under influences from European exiles after World War II. He earned his doctorate at the University of São Paulo under advisors influenced by the traditions of Eugenio Elia Levi and Gianfranco Cimmino and engaged with visiting scholars from the United States and France, including contacts linked to Courant Institute of Mathematical Sciences and École Normale Supérieure. As a young mathematician he attended conferences sponsored by organizations such as the Brazilian Mathematical Society and the International Mathematical Union, where he met contemporaries like Paul Erdős and Jean Leray.

Academic career and positions

He served on the faculty of the University of São Paulo and later at the Pontifical Catholic University of Rio de Janeiro and the Brazilian Academy of Sciences, collaborating with researchers from institutions including the Massachusetts Institute of Technology, the Institute for Advanced Study, and the University of California, Berkeley. He supervised doctoral students who went on to positions at the Universidade Federal do Rio de Janeiro and international universities tied to programs at the Max Planck Institute and the CNRS. His visiting appointments included terms at the Institute of Pure and Applied Mathematics (IMPA) and lecture series at the International Congress of Mathematicians and the American Mathematical Society.

Contributions to differential geometry

do Carmo made lasting contributions to the theory of minimal surfaces, the classification of isometric immersions, and stability questions for hypersurfaces in Euclidean space and spherical space, building on methods introduced by Jorge Santos and techniques related to the second variation and the Gauss–Bonnet theorem. He studied eigenvalue problems related to the Laplace–Beltrami operator and their geometric implications, connecting to conjectures pursued by Shing-Tung Yau and Richard Schoen. His work on the geometry of submanifolds addressed rigidity phenomena reminiscent of results by John Nash and Lars Ahlfors, and he contributed to global results that interacted with the theories advanced by Michael Atiyah, Isadore Singer, and René Thom.

His research influenced developments in the theory of curvature estimates, comparison theorems akin to those of Cheeger–Gromoll, and geometric analysis approaches connected to the Calabi conjecture and variational techniques used by Jürgen Jost and Rick Schoen. do Carmo also investigated classical surface theory topics linked to the names of Leonhard Euler and Joseph-Louis Lagrange, while integrating modern tools from the schools of Shiing-Shen Chern and Marcel Berger.

Major publications and books

He authored several widely used monographs that shaped graduate education in geometry, including texts on Riemannian geometry and the theory of submanifolds which became staples alongside works by Morris Hirsch, John Milnor, Shlomo Sternberg, and Andrew Pressley. His books addressed classical differential geometry topics treated historically by authors such as C. E. Weatherburn and more modern expositions found in series edited by the American Mathematical Society and Springer Science+Business Media. He published influential papers in journals associated with the Brazilian Mathematical Society, the Annals of Mathematics, and the Journal of Differential Geometry, often citing and extending methods from papers by Heinz Hopf and Enrico Bombieri.

Awards and honors

do Carmo received recognition from national and international bodies, including honors from the Brazilian Academy of Sciences and awards presented at meetings of the International Mathematical Union and the American Mathematical Society. He was elected to academies and societies alongside contemporaries honored by institutions such as the Royal Society and received invitations to deliver named lectures in the tradition of prizes like the Chern Medal and medals associated with the Institute of Mathematics and its Applications. His legacy is commemorated in symposia at venues including the Institute for Pure and Applied Mathematics (IMPA), the International Congress of Mathematicians, and major Brazilian universities.

Category:Brazilian mathematicians Category:Geometers Category:1928 births Category:2018 deaths