Differential geometry

Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century. Since the late 19th century, differential geometry has grown into a field concerned more with the geometric structures on differentiable manifolds. Differential geometry is related to differential topology and the geometric aspects of the theory of differential equations; the differential geometry of surfaces captures many of the key ideas and techniques endemic to this field. Differential geometry arose and developed as a result of and in connection to the mathematical analysis of curves and surfaces. Mathematical analysis of curves and surfaces had been developed to answer some of the nagging and unanswered questions that appeared in calculus, like the reasons for relationships between complex shapes and curves and analytic functions.

These unanswered questions indicated greater, hidden relationships. The general idea of natural equations for obtaining curves from local curvature appears to have been first considered by Leonhard Euler in 1736, many examples with simple behavior were studied in the 1800s; when curves, surfaces enclosed by curves, points on curves were found to be quantitatively, related by mathematical forms, the formal study of the nature of curves and surfaces became a field of study in its own right, with Monge's paper in 1795, with Gauss's publication of his article, titled'Disquisitiones Generales Circa Superficies Curvas', in Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores in 1827. Applied to the Euclidean space, further explorations led to non-Euclidean space, metric and topological spaces. Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric; this is a concept of distance expressed by means of a smooth positive definite symmetric bilinear form defined on the tangent space at each point.

Riemannian geometry generalizes Euclidean geometry to spaces that are not flat, although they still resemble the Euclidean space at each point infinitesimally, i.e. in the first order of approximation. Various concepts based on length, such as the arc length of curves, area of plane regions, volume of solids all possess natural analogues in Riemannian geometry; the notion of a directional derivative of a function from multivariable calculus is extended in Riemannian geometry to the notion of a covariant derivative of a tensor. Many concepts and techniques of analysis and differential equations have been generalized to the setting of Riemannian manifolds. A distance-preserving diffeomorphism between Riemannian manifolds is called an isometry; this notion can be defined locally, i.e. for small neighborhoods of points. Any two regular curves are locally isometric. However, the Theorema Egregium of Carl Friedrich Gauss showed that for surfaces, the existence of a local isometry imposes strong compatibility conditions on their metrics: the Gaussian curvatures at the corresponding points must be the same.

In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated with a Riemannian manifold that measures how close it is to being flat. An important class of Riemannian manifolds is the Riemannian symmetric spaces, whose curvature is not constant; these are the closest analogues to the "ordinary" plane and space considered in Euclidean and non-Euclidean geometry. Pseudo-Riemannian geometry generalizes Riemannian geometry to the case in which the metric tensor need not be positive-definite. A special case of this is a Lorentzian manifold, the mathematical basis of Einstein's general relativity theory of gravity. Finsler geometry has Finsler manifolds as the main object of study; this is a differential manifold with a Finsler metric, that is, a Banach norm defined on each tangent space. Riemannian manifolds are special cases of the more general Finsler manifolds. A Finsler structure on a manifold M is a function F: TM → [0, ∞) such that: F = m F for all in TM and all m≥0, F is infinitely differentiable in TM ∖, The vertical Hessian of F2 is positive definite.

Symplectic geometry is the study of symplectic manifolds. An symplectic manifold is a differentiable manifold equipped with a smoothly varying non-degenerate skew-symmetric bilinear form on each tangent space, i.e. a nondegenerate 2-form ω, called the symplectic form. A symplectic manifold is an symplectic manifold for which the symplectic form ω is closed: dω = 0. A diffeomorphism between two symplectic manifolds which preserves the symplectic form is called a symplectomorphism. Non-degenerate skew-symmetric bilinear forms can only exist on even-dimensional vector spaces, so symplectic manifolds have dimension. In dimension 2, a symplectic manifold is just a surface endowed with an area form and a symplectomorphism is an area-preserving diffeomorphism; the phase space of a mechanical system is a symplectic manifold and they made an implicit appearance in the work of Joseph Louis Lagrange on analytical mechanics and in Carl Gustav Jacobi's and William Rowan Hamilton's formulations of classical mechanics.

By contrast with Riemannian geometry, where the curvature provides a local invariant of Riemannian manifolds, Darboux's theorem states that all symplectic manifolds are locally isomorphic. The only invariants of a symplectic manifold are global in nature and topological aspects play a prominent role in symplectic geometry; the first result in symplectic topology is


Tandernaken, al op den Rijn was once a popular Middle Dutch song about two girls who in Andernach, a city in Germany on the left Rhine bank, were spied on by the lover of one of the girls, listening to their conversation on love affairs from a distance. The complete text of the song is preserved in the Antwerp songbook. Other versions are less complete; the tune of the song survived in monophonic and in polyphonic sources, but the text of the secular song is only known through textual sources. Tandernaken was an international hit in the period between about 1430 and the 1540s as settings, preserved in Dutch, Italian and English sources, are listed by Franco-Flemish and English composers such as Jacob Obrecht, Antoine Brumel, King Henry VIII, Alexander Agricola, Paul Hofhaimer, Petrus Alamire, Ludwig Senfl and Erasmus Lapicida; the earliest extant setting of the Tandernaken tune is by Tijling, a composer of whom, besides this composition, nothing else is known. His composition is included in one of the so-called Trent Codices.

The tenor voice has the features of the polyphonic tenores of the Dutch and French song settings from the first half of the 15th century. These same features are found in a number of tunes which were notated with lines instead of notes on a stave; these versions have all in common that the text has been noted separately from the tune or the tenor. The earliest polyphonic settings of Tandernaken are registered in Dutch or Italian sources and were by Franco-Flemish or Dutch composers; the most recent sources and compositions are found in Germany. The tune became first popular in Italy before it entering Germany by way of Italian instrumental ensembles. Most of the polyphonic settings do not give the text to the tune. Where there is a text, the text is a spiritual contrafactum. These'monophonic' sources which do not provide any musical notation include secular contrafacta. Although the text extant in the Antwerp songbook can be sung without too much difficulty by the tenor voice in the oldest settings such as these by Tijling and Obrecht, although the tune of the extant non-polyphonic versions is related to but quite different from the tenor of the polyphonic versions, most of the polyphonic compositions can be regarded as instrumental settings.

An indication of the instruments with which the non-texted polyphonic versions of Tandernaken could be played, is provided by a manuscript made for the players of wind instruments at the court of Albert of Prussia, in which the word Krumbhörner, crumhorns, is mentioned in the bass voice. A setting by Hofhaimer was notated for three voices in tablature for organ. A si-placet-altus in mensural notation was added to the tablature of Hans Kotter, with the comment von einandern darzu zuschlagen, to be performed by another player separately; the first verse of Tandernaken is included in a Dutch quodlibet. Discography on the commercial web site Discographical search results on for T’Andernaken Ulanka sings Tandernaken on Jacob Obrecht Erasmus Lapicida King Henry VIII Alexander Agricola Tune and two versions of the complete text of Tandernaken, as published by Florimond van Duyse. The first was taken from the Antwerp songbook and the second from a manuscript from the collection of the Library of the University of Amsterdam Search results for Tandernaken on the web site


Airwalker is the debut release by Jeremy Jay on K Records. The EP included two covers:'"Lunar Camel" by Siouxsie and the Banshees and "Angels on the Balcony" by Blondie; the 7 inch version contains two tracks. "Lampost Scene" – 0:40 "Airwalker" – 3:35 "We Stay Here" – 2:46 "Lunar Camel" – 2:11 "Angels on the Balcony" – 3:10 "Can We Disappear?" – 3:58 Derek Jamesbass Larissa James – photography Jeremy Jay – synthesizer, piano, producer Ilya Malinsky – guitar Nick Pahl – drums