Harry Ernest Rauch (November 9, 1925 – June 18, 1979) was an American mathematician, who worked on complex analysis and differential geometry. He was born in

Trenton, New Jersey Trenton is the capital city of the U.S. state of New Jersey and the county seat of Mercer County. It was the capital of the United States from November 1 to December 24, 1784.White Plains, New York (Always Faithful) , image_seal = WhitePlainsSeal.png , seal_link = , subdivision_type = List of sovereign states, Country , subdivision_name = , subdivision_type1 = U.S. state, State , su ...

. Rauch earned his PhD in 1948 from

Princeton University Princeton University is a private research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the ...

under

Salomon Bochner Salomon Bochner (20 August 1899 – 2 May 1982) was an Austrian mathematician, known for work in mathematical analysis, probability theory and differential geometry. Life He was born into a Jewish family in Podgórze (near Kraków), then Aus ...

with thesis ''Generalizations of Some Classic Theorems to the Case of Functions of Several Variables''. From 1949 to 1951 he was a visiting member of the

Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent schola ...

. He was in the 1960s a professor at

Yeshiva University Yeshiva University is a private Orthodox Jewish university with four campuses in New York City."About YU
on the Yeshiva Universi ...

and from the mid-1970s a professor at the City University of New York. His research was on differential geometry (especially geodesics on ''n''-dimensional manifolds),

Riemann surfaces In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ve ...

, and

theta function In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field ...

s. In the early 1950s Rauch made fundamental progress on the ''quarter-pinched sphere conjecture'' in differential geometry. In the case of positive

sectional curvature In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature ''K''(σ''p'') depends on a two-dimensional linear subspace σ''p'' of the tangent space at a poi ...

and simply connected differential manifolds, Rauch proved that, under the condition that the sectional curvature ''K'' does not deviate too much from ''K'' = 1, the manifold must be homeomorphic to the sphere (''i.e.'' the case where there is constant sectional curvature ''K'' = 1). Rauch's result created a new paradigm in differential geometry, that of a "pinching theorem;" in Rauch's case, the assumption was that the curvature was pinched between 0.76 and 1. This was later relaxed to pinching between 0.55 and 1 by Wilhelm Klingenberg, and finally replaced with the sharp result of pinching between 0.25 and 1 by

Marcel Berger Marcel Berger (14 April 1927 – 15 October 2016) was a French mathematician, doyen of French differential geometry, and a former director of the Institut des Hautes Études Scientifiques (IHÉS), France. Formerly residing in Le Castera in Las ...

and Klingenberg in the early 1960s. This optimal result is known as the sphere theorem for Riemannian manifolds. The

Rauch comparison theorem In Riemannian geometry, the Rauch comparison theorem, named after Harry Rauch, who proved it in 1951, is a fundamental result which relates the sectional curvature of a Riemannian manifold to the rate at which geodesics spread apart. Intuitivel ...

is also named after Harry Rauch. He proved it in 1951.

Publications

Articles

* * * * * with Hershel M. Farkas: * * with H. M. Farkas: * with H. M. Farkas: * with Isaac Chavel:

Books

* with Hershel M. Farkas: ''Theta functions with applications to Riemann Surfaces'', Williams and Wilkins, Baltimore 1974 * with Aaron Lebowitz: ''Elliptic functions, theta functions and Riemann Surfaces'', Williams and Wilkins, 1973 * with Matthew Graber, William Zlot: ''Elementary Geometry'', Krieger 1973, 2nd edn. 1979 * ''Geodesics and Curvature in Differential Geometry in the Large'', Yeshiva University 1959

Sources

* Hershel M. Farkas, Isaac Chavel (eds.): ''Differential geometry and complex analysis: a volume dedicated to the memory of Harry Ernest Rauch'', Springer, 1985

References

External links

* {{DEFAULTSORT:Rauch, Harry 1925 births 1979 deaths 20th-century American mathematicians Differential geometers People from Trenton, New Jersey Princeton University alumni Yeshiva University faculty City University of New York faculty 20th-century American Jews