Harry Rauch

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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, and died in White Plains, New York.

Rauch earned his PhD in 1948 from Princeton University under Salomon Bochner with thesis Generalizations of Some Classic Theorems to the Case of Functions of Several Variables.^[1] From 1949 to 1951 he was a visiting member of the Institute for Advanced Study. He was in the 1960s a professor at Yeshiva University and from the mid-1970s a professor at the Graduate School of the City University of New York. His research was on differential geometry (especially geodesics on n-dimensional manifolds), Riemann surfaces, and theta functions.

In the early 1950s Rauch made fundamental progress on the quarter-pinched sphere conjecture in differential geometry.^[2] In the case of positive sectional curvature 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 and Klingenberg in the early 1960s. This optimal result is known as the sphere theorem for Riemannian manifolds.

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