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Harry Rauch (footballer)
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''. 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 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. In the case of positive sectional curvature and simply connected differential manifolds, Rauch proved that, under the condition t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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.New Jersey County Map New Jersey Department of State. Accessed July 10, 2017. The city's metropolitan area, including all of Mercer County, is grouped with the New York combined statistical area by the |
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 point ''p'' of the manifold. It can be defined geometrically as the Gaussian curvature of the surface which has the plane σ''p'' as a tangent plane at ''p'', obtained from geodesics which start at ''p'' in the directions of σ''p'' (in other words, the image of σ''p'' under the exponential map at ''p''). The sectional curvature is a real-valued function on the 2-Grassmannian fiber bundle, bundle over the manifold. The sectional curvature determines the Riemann curvature tensor, curvature tensor completely. Definition Given a Riemannian manifold and two linearly independent tangent vectors at the same point, ''u'' and ''v'', we can define :K(u,v)= Here ''R'' is the Riemann curvature tensor, defined here by the convention R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yeshiva University Faculty
A yeshiva (; he, ישיבה, , sitting; pl. , or ) is a traditional Jewish education, Jewish educational institution focused on the study of Rabbinic literature, primarily the Talmud and halacha (Jewish law), while Torah and Jewish philosophy are studied in parallel. The studying is usually done through daily ''Shiur (Torah), shiurim'' (lectures or classes) as well as in study pairs called ''chavrusas'' (Aramaic language, Aramaic for 'friendship' or 'companionship'). ''Chavrusa''-style learning is one of the unique features of the yeshiva. In the United States and Israel, different levels of yeshiva education have different names. In the United States, elementary-school students enroll in a ''cheder'', post-Bar and Bat Mitzvah, bar mitzvah-age students learn in a ''mesivta, metivta'', and undergraduate-level students learn in a ''Beth midrash, beit midrash'' or ''yeshiva gedola'' ( he, ישיבה גדולה, , large yeshiva' or 'great yeshiva). In Israel, elementary-s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Princeton University Alumni
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 nine colonial colleges chartered before the American Revolution. It is one of the highest-ranked universities in the world. The institution moved to Newark in 1747, and then to the current site nine years later. It officially became a university in 1896 and was subsequently renamed Princeton University. It is a member of the Ivy League. The university is governed by the Trustees of Princeton University and has an endowment of $37.7 billion, the largest endowment per student in the United States. Princeton provides undergraduate and graduate instruction in the humanities, social sciences, natural sciences, and engineering to approximately 8,500 students on its main campus. It offers postgraduate degrees through the Princeton School of Publi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
People From Trenton, New Jersey
A person ( : people) is a being that has certain capacities or attributes such as reason, morality, consciousness or self-consciousness, and being a part of a culturally established form of social relations such as kinship, ownership of property, or legal responsibility. The defining features of personhood and, consequently, what makes a person count as a person, differ widely among cultures and contexts. In addition to the question of personhood, of what makes a being count as a person to begin with, there are further questions about personal identity and self: both about what makes any particular person that particular person instead of another, and about what makes a person at one time the same person as they were or will be at another time despite any intervening changes. The plural form "people" is often used to refer to an entire nation or ethnic group (as in "a people"), and this was the original meaning of the word; it subsequently acquired its use as a plural form of per ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometers
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 multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1979 Deaths
Events January * January 1 ** United Nations Secretary-General Kurt Waldheim heralds the start of the ''International Year of the Child''. Many musicians donate to the ''Music for UNICEF Concert'' fund, among them ABBA, who write the song ''Chiquitita'' to commemorate the event. ** The United States and the People's Republic of China establish full Sino-American relations, diplomatic relations. ** Following a deal agreed during 1978, France, French carmaker Peugeot completes a takeover of American manufacturer Chrysler's Chrysler Europe, European operations, which are based in United Kingdom, Britain's former Rootes Group factories, as well as the former Simca factories in France. * January 7 – Cambodian–Vietnamese War: The People's Army of Vietnam and Vietnamese-backed Kampuchean United Front for National Salvation, Cambodian insurgents announce the fall of Phnom Penh, Cambodia, and the collapse of the Pol Pot regime. Pol Pot and the Khmer Rouge retreat west to an area ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1925 Births
Nineteen or 19 may refer to: * 19 (number), the natural number following 18 and preceding 20 * one of the years 19 BC, AD 19, 1919, 2019 Films * ''19'' (film), a 2001 Japanese film * ''Nineteen'' (film), a 1987 science fiction film Music * 19 (band), a Japanese pop music duo Albums * ''19'' (Adele album), 2008 * ''19'', a 2003 album by Alsou * ''19'', a 2006 album by Evan Yo * ''19'', a 2018 album by MHD * ''19'', one half of the double album ''63/19'' by Kool A.D. * ''Number Nineteen'', a 1971 album by American jazz pianist Mal Waldron * ''XIX'' (EP), a 2019 EP by 1the9 Songs * "19" (song), a 1985 song by British musician Paul Hardcastle. * "Nineteen", a song by Bad4Good from the 1992 album '' Refugee'' * "Nineteen", a song by Karma to Burn from the 2001 album ''Almost Heathen''. * "Nineteen" (song), a 2007 song by American singer Billy Ray Cyrus. * "Nineteen", a song by Tegan and Sara from the 2007 album '' The Con''. * "XIX" (song), a 2014 song by Slip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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. Intuitively, it states that for positive curvature, geodesics tend to converge, while for negative curvature, geodesics tend to spread. The statement of the theorem involves two Riemannian manifolds, and allows to compare the infinitesimal rate at which geodesics spread apart in the two manifolds, provided that their curvature can be compared. Most of the time, one of the two manifolds is a "comparison model", generally a manifold with constant curvature , and the second one is the manifold under study : a bound (either lower or upper) on its sectional curvature is then needed in order to apply Rauch comparison theorem. Statement Let M, \widetilde be Riemannian manifolds, on which are drawn unit speed geodesic segments \gamma : , T\to M and \widetild ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere Theorem
In Riemannian geometry, the sphere theorem, also known as the quarter-pinched sphere theorem, strongly restricts the topology of manifolds admitting metrics with a particular curvature bound. The precise statement of the theorem is as follows. If ''M'' is a complete, simply-connected, ''n''-dimensional Riemannian manifold with sectional curvature taking values in the interval (1,4] then ''M'' is homeomorphic to the ''n''-sphere. (To be precise, we mean the sectional curvature of every tangent 2-plane at each point must lie in (1,4].) Another way of stating the result is that if ''M'' is not homeomorphic to the sphere, then it is impossible to put a metric on ''M'' with quarter-pinched curvature. Note that the conclusion is false if the sectional curvatures are allowed to take values in the ''closed'' interval ,4/math>. The standard counterexample is complex projective space with the Fubini–Study metric; sectional curvatures of this metric take on values between 1 and 4, w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 Lasseube, Berger was instrumental in Mikhail Gromov's accepting positions both at the University of Paris and at the IHÉS. Awards and honors *1956 Prix Peccot, Collège de France *1962 Prix Maurice Audin *1969 Prix Carrière, Académie des Sciences *1978 Prix Leconte, Académie des Sciences *1979 Prix Gaston Julia *1979–1980 President of the French Mathematical Society. *1991 Lester R. Ford Award Selected publications * Berger, M.Geometry revealed Springer, 2010. * Berger, M.: What is... a Systole? Notices of the AMS 55 (2008), no. 3, 374–376online text* * * *Berger, Marcel; Gauduchon, Paul; Mazet, Edmond: Le spectre d'une variété riemannienne. (French) Lecture Notes in Mathematics, Vol. 194 Springer-Verlag, Berlin-New York 1971. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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