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Hopf Conjecture
In mathematics, Hopf conjecture may refer to one of several conjectural statements from differential geometry and topology attributed to Heinz Hopf. Positively or negatively curved Riemannian manifolds The Hopf conjecture is an open problem in global Riemannian geometry. It goes back to questions of Heinz Hopf from 1931. A modern formulation is: : ''A compact, even-dimensional Riemannian manifold with positive sectional curvature has positive Euler characteristic. A compact, (2''d'')-dimensional Riemannian manifold with negative sectional curvature has Euler characteristic of sign (-1)^d.'' For surfaces, these statements follow from the Gauss–Bonnet theorem. For four-dimensional manifolds, this follows from the finiteness of the fundamental group and Poincaré duality and Euler–Poincaré formula equating for 4-manifolds the Euler characteristic with b_0-b_1+b_2-b_3+b_4 and Synge's theorem, assuring that the orientation cover is simply connected, so that the Betti numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
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 str ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abhandlungen Aus Dem Mathematischen Seminar Der Universität Hamburg
(English: ''Reports from the Mathematical Seminar of the University of Hamburg'') is a peer-reviewed mathematics journal published by Springer Science+Business Media. It publishes articles on pure mathematics and is scientifically coordinated by the ''Mathematisches Seminar'', an informal cooperation of mathematicians at the Universität Hamburg; its Managing Editors are Professors and Tobias Dyckerhoff. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. History The ''Abhandlungen'' were set up as a new journal by Wilhelm Blaschke in 1922 at the newly created Department of Mathematics (called ''Mathematisches Seminar'') at the newly founded Hamburgische Universität. Blaschke invited both Hermann Weyl and David Hilbert to the ''Mathematisches Seminar'' (in 1920 and 1921, respectively) to deliver talk series on their views concerning the Foundations of Mathematics. These talks formed part of the early history of the Grundlagenkrise der Mathematik and H ... [...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] |
Bad Elster
Bad Elster () is a spa town in the Vogtlandkreis district, in Saxony, Germany. It lies on the border of Bavaria and the Czech Republic in the Elster gebirge hills. It is situated on the river White Elster, and is protected from extremes of temperature by the surrounding wooded hills. It is 25 km southeast of Plauen, and 25 km northwest of Cheb. It is part of the ''Freunde im Herzen Europas'' microregion. History Elster before 1800 Two kilometers north west of the town centre lies the remains of a twelfth-century walled village, known today as the "Alte Schloss" or "Old Castle". This was first documented in 1324. In 1412 a manor was sold to the von Zedtwitz family, who held it until 1800. In 1533 the Reformation reached Adorf and its daughter church in Elster, and the first Protestant pastor was installed in 1540. The healing properties of the waters from the spring now known as the Moritzquelle were recognised well before Georg Leisner, physician to the Du ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Switzerland
; rm, citad federala, links=no). Swiss law does not designate a ''capital'' as such, but the federal parliament and government are installed in Bern, while other federal institutions, such as the federal courts, are in other cities (Bellinzona, Lausanne, Lucerne, Neuchâtel, St. Gallen a.o.). , coordinates = , largest_city = Zurich , official_languages = , englishmotto = "One for all, all for one" , religion_year = 2022 , religion_ref = , religion = , demonym = , german: link=no, Schweizer/Schweizerin, french: link=no, Suisse/Suissesse, it, svizzero/svizzera or , rm, Svizzer/Svizra , government_type = Federal assembly-independent directorial republic , leader_title1 = Federal Council , leader_name1 = , leader_title2 = , leader_name2 = Viktor Rossi , legislature = Federal Assembly , upper_house = Counci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fribourg
, Location of , Location of () () or , ; or , ; gsw, label=Swiss German, Frybùrg ; it, Friburgo or ; rm, Friburg. is the capital of the Cantons of Switzerland, Swiss canton of Canton of Fribourg, Fribourg and district of Sarine (district), La Sarine. Located on both sides of the river Saane/Sarine, on the Swiss Plateau, it is a major economic, administrative and educational centre on the cultural border between German-speaking Switzerland, German-speaking and Romandy, French-speaking Switzerland. Its Old town, Old City, one of the best-maintained in Switzerland, sits on a small rocky hill above the valley of the Sarine. In 2018, it had a population of 38,365. History Prehistory The region around Fribourg has been settled since the Neolithic period, although few remains have been found. These include some flint tools found near Bourguillon, as well as a stone hatchet and bronze tools. A river crossing was located in the area during the Roman Empire, Roman Era. The main activ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
German Mathematical Society
The German Mathematical Society (german: Deutsche Mathematiker-Vereinigung, DMV) is the main professional society of German mathematicians and represents German mathematics within the European Mathematical Society (EMS) and the International Mathematical Union (IMU). It was founded in 1890 in Bremen with the set theorist Georg Cantor as first president. Founding members included Georg Cantor, Felix Klein, Walther von Dyck, David Hilbert, Hermann Minkowski, Carl Runge, Rudolf Sturm, Hermann Schubert, and Heinrich Weber. The current president of the DMV is Ilka Agricola (2021–2022). Activities In honour of its founding president, Georg Cantor, the society awards the Cantor Medal. The DMV publishes two scientific journals, the ''Jahresbericht der DMV'' and ''Documenta Mathematica''. It also publishes a quarterly magazine for its membership the ''Mitteilungen der DMV''. The annual meeting of the DMV is called the ''Jahrestagung''; the DMV traditionally meets every four ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry & Topology
''Geometry & Topology'' is a peer-refereed, international mathematics research journal devoted to geometry and topology, and their applications. It is currently based at the University of Warwick, United Kingdom, and published by Mathematical Sciences Publishers, a nonprofit academic publishing organisation. It was founded in 1997Allyn Jackson The slow revolution of the free electronic journal Notices of the American Mathematical Society, vol. 47 (2000), no. 9, pp. 1053-1059 by a group of topologists who were dissatisfied with recent substantial rises in subscription prices of journals published by major publishing corporations. The aim was to set up a high-quality journal, capable of competing with existing journals, but with substantially lower subscription fees. The journal was open-access for its first ten years of existence and was available free to individual users, although institutions were required to pay modest subscription fees for both online access and for printed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group \text(3)). Lie groups are widely used in many parts of modern mathematics and physics. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torus Action
In algebraic geometry, a torus action on an algebraic variety is a group action of an algebraic torus on the variety. A variety equipped with an action of a torus ''T'' is called a ''T''-variety. In differential geometry, one considers an action of a real or complex torus on a manifold (or an orbifold). A normal algebraic variety with a torus acting on it in such a way that there is a dense orbit is called a toric variety (for example, orbit closures that are normal are toric varieties). Linear action of a torus A linear action of a torus can be simultaneously diagonalized, after extending the base field if necessary: if a torus ''T'' is acting on a finite-dimensional vector space ''V'', then there is a direct sum decomposition: :V = \bigoplus_ V_ where *\chi: T \to \mathbb_m is a group homomorphism, a character of ''T''. *V_ = \, ''T''-invariant subspace called the weight subspace of weight \chi. The decomposition exists because the linear action determines (and is determine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Killing Vector Field
In mathematics, a Killing vector field (often called a Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric. Killing fields are the infinitesimal generators of isometries; that is, flows generated by Killing fields are continuous isometries of the manifold. More simply, the flow generates a symmetry, in the sense that moving each point of an object the same distance in the direction of the Killing vector will not distort distances on the object. Definition Specifically, a vector field ''X'' is a Killing field if the Lie derivative with respect to ''X'' of the metric ''g'' vanishes: :\mathcal_ g = 0 \,. In terms of the Levi-Civita connection, this is :g\left(\nabla_Y X, Z\right) + g\left(Y, \nabla_Z X\right) = 0 \, for all vectors ''Y'' and ''Z''. In local coordinates, this amounts to the Killing equation :\nabla_\mu X_\nu + \nabla_ X_\mu = 0 \,. This condition is expressed ... [...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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
