Manfredo Perdigão do Carmo (15 August 1928, Maceió – 30 April 2018, Rio de Janeiro) was a Brazilian mathematician. He spent most of his career at IMPA and is seen as the doyen of

differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...

in Brazil.

Education and career

Do Carmo studied

civil engineering Civil engineering is a professional engineering discipline that deals with the design, construction, and maintenance of the physical and naturally built environment, including public works such as roads, bridges, canals, dams, airports, sewa ...

at the University of Recife from 1947 to 1951. After working a few years as engineer, he accepted a teaching position at the newly created Institute of Physics and Mathematics at Recife. On suggestion of Elon Lima, in 1959 he went to Instituto Nacional de Matemática Pura e Aplicada to improve his background and in 1960 he moved to the USA to pursue a Ph.D. in mathematics at the University of California, Berkeley under the supervision of Shiing-Shen Chern. He defended his thesis, entitled "''The Cohomology Ring of Certain Kahlerian Manifolds''", in 1963. After working again at University of Recife and at the University of Brasilia, in 1966 he became professor at Instituto Nacional de Matemática Pura e Aplicada (IMPA) in Rio de Janeiro. From 2003 to his death he was emeritus professor at the same institution. Do Carmo was a Guggenheim Fellow in

1965 Events January–February * January 14 – The Prime Minister of Northern Ireland and the Taoiseach of the Republic of Ireland meet for the first time in 43 years. * January 20 ** Lyndon B. Johnson is Second inauguration of Lyndo ...

and

1968 The year was highlighted by protests and other unrests that occurred worldwide. Events January–February * January 5 – "Prague Spring": Alexander Dubček is chosen as leader of the Communist Party of Czechoslovakia. * Januar ...

.Biography from the Guggenheim Foundation
/ref> In 1978 he was invited speaker at the

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the Nevanlinna Prize (to be rename ...

held in

Helsinki Helsinki ( or ; ; sv, Helsingfors, ) is the Capital city, capital, primate city, primate, and List of cities and towns in Finland, most populous city of Finland. Located on the shore of the Gulf of Finland, it is the seat of the region of U ...

. In 1991 he obtained a Doctorate ''

honoris causa An honorary degree is an academic degree for which a university (or other degree-awarding institution) has waived all of the usual requirements. It is also known by the Latin phrases ''honoris causa'' ("for the sake of the honour") or ''ad hono ...

'' from

Federal University of Alagoas Located in the city of Maceió, the Federal University of Alagoas ( pt, Universidade Federal de Alagoas, UFAL or Ufal) is the major university in coastal state Alagoas and one of the main research centers in Brazilian north eastern region (one o ...

and in 2012 from

University of Murcia The University of Murcia ( es, Universidad de Murcia) is the main university in Murcia, Spain. With 38,000 students, it is the largest university in the Región de Murcia. The University of Murcia is the third oldest university in Spain, after ...

and from

Federal University of Amazonas The Federal University of Amazonas ( pt, Universidade Federal do Amazonas, UFAM) is a public university located in Manaus, Amazonas, Brazil. It is the oldest university in Brazil and one of the largest universities in the northern region of B ...

. He served as president of the

Brazilian Mathematical Society The Brazilian Mathematical Society ( pt, Sociedade Brasileira de Matemática, SBM) is a professional association founded in 1969 at Instituto de Matemática Pura e Aplicada to promote mathematics education in Brazil. Presidents *1969–1971 ...

in the term 1971–1973. He was elected a member of the

Brazilian Academy of Sciences The Brazilian Academy of Sciences ( pt, italic=yes, Academia Brasileira de Ciências or ''ABC'') is the national academy of Brazil. It is headquartered in the city of Rio de Janeiro and was founded on May 3, 1916. Publications It publishes a lar ...

in 1970, a member of

The World Academy of Sciences The World Academy of Sciences (TWAS) is a merit-based science academy established for developing countries, uniting 1,000 scientists in some 70 countries. Its principal aim is to promote scientific capacity and excellence for sustainable deve ...

(TWAS) in 1997 and a fellow of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings ...

In 2013. Among his awards, he received the Prêmio Almirante Álavaro Alberto from the

National Council for Scientific and Technological Development The Brazilian National Council for Scientific and Technological Development (CNPq, pt, Conselho Nacional de Desenvolvimento Científico e Tecnológico, earlier ) is an organization of the Brazilian federal government under the Ministry of Scien ...

in 1984, the TWAS Prize in Mathematics in 1992, the

National Order of Scientific Merit The National Order of Scientific Merit ( pt, Ordem Nacional do Mérito Científico) is an honor bestowed upon Brazilian and foreign personalities recognized for their scientific and technical contributions to the cause and development of science in ...

in 1995 and the ''Comenda Graciliano Ramos'' from the municipality of Maceió in 2000. Do Carmo died on 30 April 2018 at the age of 89.

Research

Do Carmo's main research interests were

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to po ...

and the

differential geometry of surfaces In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectiv ...

. In particular, he worked on rigidity and convexity of isometric immersions, stability of

hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Eucl ...

s and of

minimal surface In mathematics, a minimal surface is a surface that locally minimizes its area. This is equivalent to having zero mean curvature (see definitions below). The term "minimal surface" is used because these surfaces originally arose as surfaces tha ...

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ho ...

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...

isoperimetric problem In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n b ...

s, minimal submanifolds of a

sphere A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

, and manifolds of constant

mean curvature In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. T ...

and vanishing

scalar curvature In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry ...

. Do Carmo published more than 100 papers in peer-reviewed journals; in 2012 a selection of his works was published by

Springer Springer or springers may refer to: Publishers * Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag. ** Springer Nature, a multinationa ...

. He is also known for his textbooks: they were translated into many languages and used in courses from universities such as Harvard and

Columbia Columbia may refer to: * Columbia (personification), the historical female national personification of the United States, and a poetic name for America Places North America Natural features * Columbia Plateau, a geologic and geographic region in ...

. He supervised 27 PhD students, including Celso Costa,

Marcos Dajczer Marcos Dajczer (born 19 November 1948, in Buenos Aires) is an Argentine-born Brazilian mathematician whose research concerns geometry and topology. Dajczer obtained his Ph.D. from the Instituto Nacional de Matemática Pura e Aplicada in 1980 ...

and

Keti Tenenblat Keti Tenenblat (born 27 November 1944 in Izmir, Turkey) is a Turkish- Brazilian mathematician working on Riemannian geometry, the applications of differential geometry to partial differential equations, and Finsler geometry. Together with Chuu- ...

Books

*''Differential Geometry of Curves and Surfaces'',

Prentice-Hall Prentice Hall was an American major educational publisher owned by Savvas Learning Company. Prentice Hall publishes print and digital content for the 6–12 and higher-education market, and distributes its technical titles through the Safari B ...

, 1976 *''Riemannian Geometry'', Birkhäuser, 1992 *''Differential Forms and Applications'', Springer Verlag, Universitext, 1994 *(with Eduardo Wagner and Augusto Cezar de Oliveira Morgado). ''Trigonometria – Números Complexos''

References

{{DEFAULTSORT:Carmo, Manfredo do 1928 births 2018 deaths People from Maceió Differential geometers 20th-century Brazilian mathematicians University of California, Berkeley alumni Instituto Nacional de Matemática Pura e Aplicada researchers Fellows of the American Mathematical Society Members of the Brazilian Academy of Sciences Brazilian Mathematical Society TWAS laureates Textbook writers