The cause was a brain hemorrhage, said his son Hans Christian.
Dr. Gromoll's research, including a fancifully named "soul theorem," formed some of the groundwork leading to the proof, in 2003, of the Poincaré Conjecture, one of the most famous and intractable problems in mathematics, by Grigori Perelman, a Russian mathematician. (The conjecture essentially says that any shape that does not have any holes and that fits within a finite space can be stretched and deformed into a sphere -- although Henri Poincaré was conjecturing about shapes and spheres of a higher dimension.)
Bruce Kleiner, a professor of mathematics at Yale, said that Dr. Gromoll and Jeff Cheeger, a mathematician now at New York University, had provided "one of the cornerstones" of the Poincaré Conjecture solution. "And it actually shows up in two independent places in Perelman's proof," Dr. Kleiner said, "and is a really essential ingredient in that work."
In work from the 1960s and 1970s, Dr. Gromoll and his collaborators looked at how knowledge about the local bending of a surface could provide information about the surface's global structure. "Geometry became one of the most active and important areas" of mathematics, said H. Blaine Lawson, a colleague of Dr. Gromoll's at Stony Brook University, where Dr. Gromoll was a professor of mathematics. "Detlef is certainly one of the main people responsible for that."
In the soul theorem, published in 1972, Dr. Gromoll and Dr. Cheeger were studying the properties of certain surfaces that could have flat regions or curves like the outside of a sphere but not regions shaped liked saddles. They found that the properties of such surfaces, infinite in extent and existing in any number of dimensions, could be deduced from a finite central core region.
Dr. Cheeger said it was Dr. Gromoll who suggested calling this finite region the "soul" of the object, because it captured the essence of the infinite expanse around it. "Just like inside a person," Dr. Cheeger said.
Dr. Lawson called it "a phenomenally beautiful theorem."
Detlef Gromoll was born in Berlin in 1938, and his childhood was disrupted by the falling bombs of World War II. While his father, an electrical engineer, was away developing systems for the landing of airplanes in bad weather, he and his mother fled Berlin and became refugees, moving from village to village before settling in Rosdorf, in northern Germany.
Dr. Gromoll's college education at the University of Bonn culminated with a doctoral degree in mathematics in 1964. After moving among several universities -- Princeton, Mainz University in Germany, the University of California, Berkeley, and the University of Bonn -- Dr. Gromoll joined the State University of New York at Stony Brook in 1969.
In addition to his son Christian, who lives in Charlottesville, Va., Dr. Gromoll is survived by his wife, Suzan; another son, Stefan, of Setauket, N.Y.; and a daughter, Heidi, of Detroit.
Dr. Cheeger recalled that Dr. Gromoll was a perfectionist in his approach to home repairs as well as to mathematics. "He had taken something like the toilet apart and it would remain in that condition for the next two years," Dr. Cheeger said. "He wanted to do something and he didn't care how long it took and he wanted it to be done right."
In mathematical collaborations, Dr. Cheeger said, Dr. Gromoll would discuss the wording of every sentence in a paper he was writing. "It was a lot of fun," Dr. Cheeger said, "but it could be maddening."
By KENNETH CHANG
Published: June 19, 2008 © New York Times