# Harry Rauch

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**Harry Ernest Rauch** (November 9, 1925 – June 18, 1979) was an American mathematician, who worked on [complex analysis](/source/Complex_analysis) and [differential geometry](/source/Differential_geometry). He was born in [Trenton, New Jersey](/source/Trenton,_New_Jersey), and died in [White Plains, New York](/source/White_Plains,_New_York).

Rauch earned his PhD in 1948 from [Princeton University](/source/Princeton_University) under [Salomon Bochner](/source/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](/source/Institute_for_Advanced_Study). He was in the 1960s a professor at [Yeshiva University](/source/Yeshiva_University) and from the mid-1970s a professor at the [Graduate School of the City University of New York](/source/Graduate_School_of_the_City_University_of_New_York). His research was on differential geometry (especially [geodesics](/source/Geodesic) on *n*-dimensional manifolds), [Riemann surfaces](/source/Riemann_surfaces), and [theta functions](/source/Theta_function).

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](/source/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](/source/Wilhelm_Klingenberg), and finally replaced with the sharp result of pinching between 0.25 and 1 by [Marcel Berger](/source/Marcel_Berger) and Klingenberg in the early 1960s. This optimal result is known as the [sphere theorem for Riemannian manifolds](/source/Sphere_theorem).

The [Rauch comparison theorem](/source/Rauch_comparison_theorem) is also named after Harry Rauch. He proved it in 1951.

## Publications

### Articles

- Rauch, H. E. (1951). "A contribution to differential geometry in the large". *[Annals of Mathematics](/source/Annals_of_Mathematics)*. **54** (1): 38–55. [doi:10.2307/1969309](https://doi.org/10.2307/1969309). [JSTOR 1969309](https://www.jstor.org/stable/1969309). MR 42765.
- Rauch, H. E. (1962). ["The singularities of the modulus space"](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-68/issue-4/The-singularities-of-the-modulus-space/bams/1183524682.pdf). *[Bulletin of the American Mathematical Society](/source/Bulletin_of_the_American_Mathematical_Society)*. **68** (4): 390–394. [doi:10.1090/s0002-9904-1962-10818-0](https://doi.org/10.1090/s0002-9904-1962-10818-0). MR 0141781.
- Rauch, H. E. (1965). "A transcendental view of the space of algebraic Riemann surfaces". *Bulletin of the American Mathematical Society*. **71** (1): 1–39. [doi:10.1090/s0002-9904-1965-11225-3](https://doi.org/10.1090/s0002-9904-1965-11225-3). MR 0213543.
- Rauch, H. E. (1967). ["The local ring of the genus three modulus space of Klein's 168 surface"](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-73/issue-3/The-local-ring-of-the-genus-three-modulus-space-at/bams/1183528837.pdf). *Bulletin of the American Mathematical Society*. **73** (3): 343–346. [doi:10.1090/s0002-9904-1967-11743-9](https://doi.org/10.1090/s0002-9904-1967-11743-9). MR 0213545.
- with Hershel M. Farkas: Rauch, H. E. & Farkas, H. M. (1968). "Relation between two kinds of theta constants on a Riemann surface". *[Proceedings of the National Academy of Sciences of the United States of America](/source/Proceedings_of_the_National_Academy_of_Sciences_of_the_United_States_of_America)*. **59** (1): 52–55. [Bibcode:1968PNAS...59...52R](https://ui.adsabs.harvard.edu/abs/1968PNAS...59...52R). [doi:10.1073/pnas.59.1.52](https://doi.org/10.1073/pnas.59.1.52). [PMC 285999](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC285999). [PMID 16591592](https://pubmed.ncbi.nlm.nih.gov/16591592)
- Rauch, H. E. (1968). "Functional independence of theta constants". *Bulletin of the American Mathematical Society*. **74** (4): 633–638. [doi:10.1090/s0002-9904-1968-11969-x](https://doi.org/10.1090/s0002-9904-1968-11969-x). MR 0226000.
- with H. M. Farkas: Farkas, H. M. & Rauch, H. E. (1969). "Two kinds of theta constants and period relations on a Riemann surface". *Proceedings of the National Academy of Sciences of the United States of America*. **62** (3): 679–686. [Bibcode:1969PNAS...62..679F](https://ui.adsabs.harvard.edu/abs/1969PNAS...62..679F). [doi:10.1073/pnas.62.3.679](https://doi.org/10.1073/pnas.62.3.679). [PMC 223651](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC223651). [PMID 16591737](https://pubmed.ncbi.nlm.nih.gov/16591737)
- with H. M. Farkas: Farkas, Hershel M. & Rauch, Harry E. (1970). "Period relations of Schottky type on Riemann surfaces". *Annals of Mathematics*. **92** (2): 434–461. [doi:10.2307/1970627](https://doi.org/10.2307/1970627). [JSTOR 1970627](https://www.jstor.org/stable/1970627). MR 0283193.
- with Isaac Chavel: Chavel, I & Rauch, H. E. (1972). "Holomorphic embedding of complex curves in spaces of constant holomorphic curvature". *Proceedings of the National Academy of Sciences of the United States of America*. **69** (3): 663–665. [Bibcode:1972PNAS...69..633C](https://ui.adsabs.harvard.edu/abs/1972PNAS...69..633C). [doi:10.1073/pnas.69.3.633](https://doi.org/10.1073/pnas.69.3.633). [PMC 426523](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC426523). [PMID 16591967](https://pubmed.ncbi.nlm.nih.gov/16591967)

### 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

1. Abresch, Uwe & Meyer, Wolfgang T. (1997). ["Injectivity radius estimates and sphere theorems"](https://web.archive.org/web/20170809043344/http://library.msri.org/books/Book30/files/abresch.pdf). *Comparison Geometry*. **30**: 1 47. MSRI Publications. Archived from [the original](http://library.msri.org/books/Book30/files/abresch.pdf) on 2017-08-09. Retrieved 2012-09-22.

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