Projective differential geometry

Last updated May 03, 2019

In mathematics, projective differential geometry is the study of differential geometry, from the point of view of properties of mathematical objects such as functions, diffeomorphisms, and submanifolds, that are invariant under transformations of the projective group. This is a mixture of the approaches from Riemannian geometry of studying invariances, and of the Erlangen program of characterizing geometries according to their group symmetries.

Mathematics Field of study concerning quantity, patterns and change

Mathematics includes the study of such topics as quantity, structure, space, and change.

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.

In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable. The concept of function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.

The area was much studied by mathematicians from around 1890 for a generation (by J. G. Darboux, George Henri Halphen, Ernest Julius Wilczynski, E. Bompiani, G. Fubini, Eduard Čech, amongst others), without a comprehensive theory of differential invariants emerging. Élie Cartan formulated the idea of a general projective connection, as part of his method of moving frames; abstractly speaking, this is the level of generality at which the Erlangen program can be reconciled with differential geometry, while it also develops the oldest part of the theory (for the projective line), namely the Schwarzian derivative, the simplest projective differential invariant.^[1]

Ernest Julius Wilczynski was an American mathematician considered the founder of projective differential geometry.

Eduard Čech was a Czech mathematician born in Stračov. His research interests included projective differential geometry and topology. He is especially known for the technique known as Stone–Čech compactification and the notion of Čech cohomology.

In mathematics, a differential invariant is an invariant for the action of a Lie group on a space that involves the derivatives of graphs of functions in the space. Differential invariants are fundamental in projective differential geometry, and the curvature is often studied from this point of view. Differential invariants were introduced in special cases by Sophus Lie in the early 1880s and studied by Georges Henri Halphen at the same time. Lie (1884) was the first general work on differential invariants, and established the relationship between differential invariants, invariant differential equations, and invariant differential operators.

Further work from the 1930s onwards was carried out by J. Kanitani, Shiing-Shen Chern, A. P. Norden, G. Bol, S. P. Finikov and G. F. Laptev. Even the basic results on osculation of curves, a manifestly projective-invariant topic, lack any comprehensive theory. The ideas of projective differential geometry recur in mathematics and its applications, but the formulations given are still rooted in the language of the early twentieth century.

Shiing-Shen Chern was a Chinese-American mathematician who made fundamental contributions to differential geometry and topology. He was widely regarded as a leader in geometry and one of the greatest mathematicians of the twentieth century, winning numerous awards and recognition including the Wolf Prize and the inaugural Shaw Prize.

Curve object similar to a line but which is not required to be straight

In mathematics, a curve is, generally speaking, an object similar to a line but that need not be straight. Thus, a curve is a generalization of a line, in that it may be curved.

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In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

Christian Felix Klein was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group theory. His 1872 Erlangen Program, classifying geometries by their basic symmetry groups, was an influential synthesis of much of the mathematics of the time.

The Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as Vergleichende Betrachtungen über neuere geometrische Forschungen. It is named after the University Erlangen-Nürnberg, where Klein worked.

In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are various kinds of connections in modern geometry, depending on what sort of data one wants to transport. For instance, an affine connection, the most elementary type of connection, gives a means for transporting tangent vectors to a manifold from one point to another along a curve. An affine connection is typically given in the form of a covariant derivative, which gives a means for taking directional derivatives of vector fields: the infinitesimal transport of a vector field in a given direction.

Homogeneous space Topological space in group theory

In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively. The elements of G are called the symmetries of X. A special case of this is when the group G in question is the automorphism group of the space X – here "automorphism group" can mean isometry group, diffeomorphism group, or homeomorphism group. In this case, X is homogeneous if intuitively X looks locally the same at each point, either in the sense of isometry, diffeomorphism, or homeomorphism (topology). Some authors insist that the action of G be faithful, although the present article does not. Thus there is a group action of G on X which can be thought of as preserving some "geometric structure" on X, and making X into a single G-orbit.

In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form. Cartan connections describe the geometry of manifolds modelled on homogeneous spaces.

In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.

In mathematics, the Schwarzian derivative, named after the German mathematician Hermann Schwarz, is a certain operator that is invariant under all Möbius transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms and hypergeometric functions. It plays an important role in the theory of univalent functions, conformal mapping and Teichmüller spaces.

In mathematics, a differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.

In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. These objects are typically functions on $, functions on a manifold, vector valued functions, vector fields, or, more generally, sections of a vector bundle.$

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.

This is a glossary of terms that are or have been considered areas of study in mathematics.

Robert Hermann is an American mathematician and mathematical physicist. In the 1960s Hermann worked on elementary particle physics and quantum field theory, and published books which revealed the interconnections between vector bundles on Riemannian manifolds and gauge theory in physics, before these interconnections became "common knowledge" among physicists in the 1970s.

Sergei Tabachnikov, also spelled Serge, is a Russian mathematician who works in geometry and dynamical systems. He is currently a Professor of Mathematics at Pennsylvania State University.

Ellis Bagley Stouffer was an American mathematician specializing in projective differential geometry.

Ernest Preston Lane was an American mathematician, specializing in differential geometry.

References

↑ V. Ovsienko and S. Tabachnikov (2004). Projective Differential Geometry Old and New From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups (PDF). Cambridge University Press. p. vii (preface). ISBN 9780521831864.

Ernest Julius Wilczynski Projective differential geometry of curves and ruled surfaces (Leipzig: B.G. Teubner,1906)

Projective differential geometry

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