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Riemannian Connection On A Surface
In mathematics, the Riemannian connection on a surface or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early part of the twentieth century: parallel transport, covariant derivative and connection form. These concepts were put in their current form with principal bundles only in the 1950s. The classical nineteenth century approach to the differential geometry of surfaces, due in large part to Carl Friedrich Gauss, has been reworked in this modern framework, which provides the natural setting for the classical theory of the moving frame as well as the Riemannian geometry of higher-dimensional Riemannian manifolds. This account is intended as an introduction to the theory of connections. Historical overview After the classical work of Gauss on the differential geometry of surfaces and the subsequent emergence of the concept of Riemannian manifold initiated by Bernhard Riemann in the mi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is associated with the University of Göttingen tradition of mathematics, represented by Carl Friedrich Gauss, David Hilbert and Hermann Minkowski. His research has had major significance for theoretical physics as well as purely mathematical disciplines such as number theory. He was one of the most influential mathematicians of the twentieth century, and an important member of the Institute for Advanced Study during its early years. Weyl contributed to an exceptionally wide range of mathematical fields, including works on space, time, matter, philosophy, logic, symmetry and the history of mathematics. He was one of the first to conceive of combining general relativity with the laws of electromagnetism. Freeman Dyson wrote that Weyl alone bore co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science). This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative. Definition An order-m linear differential operator is a map A from a function space \mathcal_1 to another function space \mathcal_2 that can be written as: A = \sum_a_\alpha(x) D^\alpha\ , where \alpha = (\alpha_1,\alpha_2,\cdots,\alpha_n) is a multi-index of non-negative integers, , \alpha, = \alpha_1 + \alpha_2 + \cdots + \alpha_n, and for each \alpha, a_\alpha(x) is a function on some open domain in ''n''-dimensional space. The operator D^\alpha is interpreted as D^\alp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levi-Civita Connection
In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves the (pseudo-)Riemannian metric and is torsion-free. The fundamental theorem of Riemannian geometry states that there is a unique connection which satisfies these properties. In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection. The components (structure coefficients) of this connection with respect to a system of local coordinates are called Christoffel symbols. History The Levi-Civita connection is named after Tullio Levi-Civita, although originally "discovered" by Elwin Bruno Christoffel. Levi-Civita, along with Gregorio Ricci-Curbastro, used Christoffel's symbols to define the notion of parallel transport and explore the relationship of parallel transp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Katsumi Nomizu
was a Japanese-American mathematician known for his work in differential geometry. Life and career Nomizu was born in Osaka, Japan on the first day of December, 1924. He studied mathematics at Osaka University, graduating in 1947 with a Master of Science then traveled to the United States on a U.S. Army Fulbright Scholarship. He studied first at Columbia University and then at the University of Chicago where in 1953 he became the first student to earn a Ph.D. under the thesis direction of Shiing-Shen Chern. The subject was affine differential geometry, a topic to which he would return much later in his career. He presented his thesis, ''Invariant affine connections on homogeneous spaces'' in 1953. Returning to Japan, he studied at Nagoya University, obtaining a doctor of science in 1955. He published his first volume, ''Lie Groups and Differential Geometry'' dedicated to his wife Kimiko whom he had married that same year. Nomizu taught at Nagoya University until 1958 when ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shoshichi Kobayashi
was a Japanese mathematician. He was the eldest brother of electrical engineer and computer scientist Hisashi Kobayashi. His research interests were in Riemannian and complex manifolds, transformation groups of geometric structures, and Lie algebras. Biography Kobayashi graduated from the University of Tokyo in 1953. In 1956, he earned a Ph.D. from the University of Washington under Carl B. Allendoerfer. His dissertation was ''Theory of Connections''. He then spent two years at the Institute for Advanced Study and two years at MIT. He joined the faculty of the University of California, Berkeley in 1962 as an assistant professor, was awarded tenure the following year, and was promoted to full professor in 1966. Kobayashi served as chairman of the Berkeley Mathematics Department for a three-year term from 1978 to 1981 and for the 1992 Fall semester. He chose early retirement under the VERIP plan in 1994. The two-volume book ''Foundations of Differential Geometry'', which he coau ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isadore M
Isidore ( ; also spelled Isador, Isadore and Isidor) is an English and French masculine given name. The name is derived from the Greek name ''Isídōros'' (Ἰσίδωρος) and can literally be translated to "gift of Isis." The name has survived in various forms throughout the centuries. Although it has never been a common name, it has historically been popular due to its association with Catholic figures and among the Jewish diaspora. Isidora is the feminine form of the name. Pre-modern era :''Ordered chronologically'' Religious figures * Isidore of Alexandria (died 403), Egyptian priest, saint * Isidore of Chios (died 251), Roman Christian martyr * Isidore of Scété (died c. 390), 4th-century A.D. Egyptian Christian priest and desert ascetic * Isidore of Pelusium (died c. 449), Egyptian monk, saint and prolific letter writer * Isidore of Seville (c. 560–636), Catholic saint and scholar, last of the Fathers of the Church and Archbishop of Seville * Isidore the Laborer (c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Ehresmann
Charles Ehresmann (19 April 1905 – 22 September 1979) was a German-born French mathematician who worked in differential topology and category theory. He was an early member of the Bourbaki group, and is known for his work on the differential geometry of smooth fiber bundles, notably the introduction of the concepts of Ehresmann connection and of jet bundles, and for his seminar on category theory. Life Ehresmann was born in Strasbourg (at the time part of the German Empire) to an Alsatian-speaking family; his father was a gardener. After World War I, Alsace returned part of France and Ehresmann was taught in French at Lycée Kléber. Between 1924 and 1927 he studied at the École Normale Supérieure (ENS) in Paris and obtained agrégation in mathematics. After one year of military service, in 1928-29 he taught at a French school in Rabat, Morocco. He studied further at the University of Göttingen during the years 1930–31, and at Princeton University in 1932–34. He co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levi-Civita
Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made significant contributions in other areas. He was a pupil of Gregorio Ricci-Curbastro, the inventor of tensor calculus. His work included foundational papers in both pure and applied mathematics, celestial mechanics (notably on the three-body problem), analytic mechanics (the Levi-Civita separability conditions in the Hamilton–Jacobi equation) and hydrodynamics. Biography Born into an Italian Jewish family in Padua, Levi-Civita was the son of Giacomo Levi-Civita, a lawyer and former senator. He graduated in 1892 from the University of Padua Faculty of Mathematics. In 1894 he earned a teaching diploma after which he was appointed to the Faculty of Science teacher's college in Pavia. In 1898 he was appointed to the Padua Chair of Rational Mechanics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ricci
Ricci () is an Italian surname, derived from the adjective "riccio", meaning curly. Notable Riccis Arts and entertainment * Antonio Ricci (painter) (c.1565–c.1635), Spanish Baroque painter of Italian origin * Christina Ricci (born 1980), American actress * Federico Ricci (1809–77), Italian composer * Franco Maria Ricci (1937–2020), Italian art publisher * Italia Ricci (born 1986), Canadian actress * Jason Ricci (born 1974), American blues harmonica player * Lella Ricci (1850–71), Italian singer * Luigi Ricci (1805–59), Italian composer * Luigi Ricci (1893–1981), Italian vocal coach * Luigi Ricci-Stolz (1852–1906), Italian composer * Marco Ricci (1676–1730), Italian Baroque painter * Nahéma Ricci, Canadian actress * Nina Ricci (designer) (1883–1970), French fashion designer * Nino Ricci (born 1959), Canadian novelist * Regolo Ricci (born 1955), Canadian painter and illustrator * Ruggiero Ricci (1918–2012), American violinist * Sebastiano Ricci (1659–1734), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chern–Weil Homomorphism
In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold ''M'' in terms of connections and curvature representing classes in the de Rham cohomology rings of ''M''. That is, the theory forms a bridge between the areas of algebraic topology and differential geometry. It was developed in the late 1940s by Shiing-Shen Chern and André Weil, in the wake of proofs of the generalized Gauss–Bonnet theorem. This theory was an important step in the theory of characteristic classes. Let ''G'' be a real or complex Lie group with Lie algebra and let \Complexmathfrak g/math> denote the algebra of \Complex-valued polynomials on \mathfrak g (exactly the same argument works if we used \R instead of Let \Complexmathfrak gG be the subalgebra of fixed points in \Complexmathfrak g/math> under the adjoint action of ''G''; that is, the subalgebra consisting of all poly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
