Geometry

The branch of mathematics whose primary subject is spatial relationships and shapes of bodies. Geometry studies spatial relationships and shapes, while ignoring other properties of real bodies (density, weight, colour, etc.). In its subsequent stages of development geometry also dealt with other relations and shapes of reality, similar to spatial shapes. In our own times geometry is generally understood to study any relationships or shapes that appear in the study of homogeneous objects, phenomena, events — apart from their physical contents — which resemble ordinary spatial relations and shapes. For instance, it considers distances between functions irrespective of any special properties of these functions and of any real processes described by these functions (see, e.g., Metric space; Functional analysis).

Historical review.

Geometry was born in early Antiquity. Its birth was due to practical necessities (measurements of parts of the Earth, determinations of volumes of solid bodies). The simplest geometrical concepts and postulates were already known to early Egyptians and Babylonians (beginning of the second millennium B.C.). In those days geometric postulates were formulated as rules, with primitive logical proofs or without any proofs at all. During the period between the 7th century B.C. and the 1st century A.D. geometry mainly developed in Ancient Greece. The result was an accumulation of knowledge on metric relationships governing triangles, determinations of areas and volumes, proportions and similitudes of figures, conical sections and construction problems. Relatively rigorous logical proofs of geometric theorems also appeared during that period. The $ Elements $ of Euclid was a compendium of known facts in geometry and their systematization (about 300 B.C.). This work comprises the formulation of the fundamental assumptions (postulates) of geometry, from which various properties of the simplest planar and spatial figures could be deduced by logical reasoning. It also laid the first foundations of the axiomatic method. The development of astronomy and geodesy in the first century and 2nd century A.D. resulted in the advent of plane and three-dimensional trigonometry.

Subsequent development of geometry, up to the 17th century, was less intensive. The rebirth of arts and sciences in Europe favoured the development of geometry. The theory of perspective, dealing with the representation of solid bodies on a plane surface (cf. Descriptive geometry) was in the centre of interest of artists and architects. These needs resulted in the advent of projective geometry — the branch of geometry dealing with the properties of figures that are invariant under so-called projective transformations.

An altogether novel approach to geometric problems was proposed in the early 17th century by R. Descartes (1596–1650). He created the method of coordinates, which made it possible to use algebraic methods, and, at a later date, to use analysis, in geometry. This marks the beginning of an intensive development in geometry. Analytic geometry, in which the properties of curves and surfaces defined by algebraic equations are studied by algebraic methods, arises. The use of analytical methods in geometry by L. Euler (1707–1783) and G. Monge (1746–1818) in the 18th century laid the foundations of classical differential geometry. Its principal parts are the theories of curves and surfaces, and they were intensively developed and generalized by C.F. Gauss (1777–1855) and other geometers. As a result of the interaction of geometry with algebra and analysis there followed the appearance of special calculi, which are conveniently used in geometry and in other fields of mathematics (Vector calculus; Tensor calculus; the method of differential forms, cf. Differential form).

That part of geometry not based on the methods of algebra and analysis, and operating directly with geometric images, received the name synthetic geometry.

The subject of geometry, its principal branches and connection with other branches of mathematics.

The first steps in geometry were made from the aspect of physical science, and the first results obtained concerned most properties of physically-observable magnitudes. Subsequently, up to the second half of the 19th century, geometry dealt with the relationships and shapes of bodies in space, the properties of which were determined by the axioms formulated by Euclid (cf. Euclidean geometry). Euclidean space gives a description of the simplest physical observations which is so satisfactory that, up to the 19th century, it was, for all practical purposes, identified with physical space. N.I. Lobachevskii (1793–1856) in 1826 constructed a geometry (cf. Lobachevskii geometry) based on a set of axioms that differed from Euclid's only by the axiom on parallel straight lines (cf. Fifth postulate). The result was a logically-consistent geometry that substantially differed from Euclidean geometry. It became clear that different spaces with inherent geometries can be constructed in mathematics (see, for example, Non-Euclidean geometries). This was also the genesis of the idea of a multi-dimensional space. The subsequent novel step in geometry was an idea of B. Riemann (1826–1866), who in 1854 formulated a generalized concept of space as a continuous family of arbitrary homogeneous objects or phenomena, and who introduced spaces in which distances are measured according to some given law of "infinitesimal steps" (a metric is introduced). In other words, one specifies some function that expresses the distances along a curve by differentials in the coordinates corresponding to small displacements. The development of this idea of Riemann subsequently yielded various generalized methods for specifying the metrics and led to studies of the geometry of the spaces thus obtained (cf. Riemannian space; Finsler space). In studying physical spaces of various mechanical systems, or systems of uniform physical objects in general, the choice of a suitable mathematical space, and bringing its elements into correspondence with the objects of the system under study, depends on the nature of the system in question. The quality of such a mathematical modelling is verified by experiments. Various objects, as well as identical objects investigated in greater or lesser detail, may require the use of different spaces. In the general physical theory of space-time (cf. Relativity theory) one kind of Riemannian space is employed.

One of the stimuli in the development and systematization of geometry was its connection with group theory. F. Klein (1849–1925) in 1872 defined the thematic content of geometry in the Erlangen program as follows: A manifold and a group of transformations on it are given. One is required to develop the theory of invariants of this group. For instance, the theory of invariants of an orthogonal group defines Euclidean geometry. Such a classification is also satisfactory for affine geometry; conformal geometry and projective geometry. However, Riemannian geometry cannot be defined in this manner. E. Cartan accordingly introduced spaces in which the corresponding group of transformations has a local action only, in an infinitesimal neighbourhood; this applies to Riemannian spaces and to spaces with different connections (cf. Connection). S. Lie proposed a group approach from the point of view of groups of continuous transformations.

The logical analysis of the foundations of geometry was conducted in parallel, towards the end of the 19th century. D. Hilbert (1862–1945) in 1899 summarized in his book Grundlagen der Geometrie (cf. Foundations of geometry) the consistency, minimality and completeness of the system of axioms of geometry.

The modern conception of space as a continuous family of homogeneous objects (phenomena, states, figures, functions) is due to the fact that geometry is closely connected with other branches of mathematics. This connection was very forcefully manifested during the development of geometry in the 20th century, when geometry became strongly differentiated and its limits became less sharp as a result of the unity of mathematics as a whole. In our days a space in mathematics is conceived of as a set having a certain structure, i.e. with certain relations between its elements or subsets. The study of the simplest fairly general structure that may be called continuous resulted in the development out of geometry of the large independent branch of mathematics called topology (cf. Topology, general). Geometry assumes the existence of richer structures. With the use of analytic tools, complementary structures (connections, metrics, conformal and symplectic structures, etc.) are usually specified with the aid of tensor (in particular, vector) or other fields.

The study of several geometric structures can also concern other fields of mathematics. This is connected with the method of study employed. Thus, algebraic geometry studies algebraic varieties and related algebraic and arithmetical problems. The algebraization of geometric laws makes it possible to construct geometries over arbitrary fields (including finite fields — finite geometries). These branches are a part of algebra. Infinite-dimensional spaces are studied in functional analysis. However, in all these branches of mathematics the geometric way of reasoning remains useful, involving as it does direct operations on images, without recourse to computations.

The most traditional subject of geometry still are spaces that are manifolds with a certain supplementary structure, manifolds of different shapes — in particular their submanifolds and fields of objects of different kinds on the manifolds. Many branches of geometry can be characterized by the type of their spaces and of the objects under study in them. Thus, the global geometry of differentiable manifolds is the study of a manifold with smooth structure, of smooth manifolds and smooth fields on them. Moreover, it studies them in the large, on the complete manifolds. Geometry in the large studies similar problems for curves and surfaces with allowance for non-smoothness and singularities; it originates from the theory of convex bodies (cf. Convex body), founded by H. Minkowski (1864–1909). In integral geometry measures on samples of geometric objects are studied. Combinatorial geometry studies the combinatorial structure of geometric objects by means of topological and metric tools (e.g. densest packings and least coverings) in Euclidean, hyperbolic and elliptic spaces of various dimensions.

The development and applications of geometry and the development of the geometric perception of abstract objects in various branches of mathematics and science testify to the importance of geometry as one of the most powerful and fruitful sources of ideas and methods for investigating the real world.

See also the references to individual geometric disciplines.

References

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[2a]	A.D. Aleksandrov, "A general view of mathematics" A.D. Aleksandrov (ed.) A.N. Kolmogorov (ed.) M.A. Lavrent'ev (ed.) , Mathematics, its content, methods and meaning , 1 , M.I.T. (1969) pp. 1–64 (Translated from Russian) MR0171675 Zbl 0111.00103
[2b]	B.N. Delone, "Analytic geometry" A.D. Aleksandrov (ed.) A.N. Kolmogorov (ed.) M.A. Lavrent'ev (ed.) , Mathematics, its content, methods and meaning , 1 , M.I.T. (1969) pp. 183–260 (Translated from Russian)
[2c]	A.D. Aleksandrov, "Non-euclidean geometry" A.D. Aleksandrov (ed.) A.N. Kolmogorov (ed.) M.A. Lavrent'ev (ed.) , Mathematics, its content, methods and meaning , 2 , M.I.T. (1969) pp. 97–189 (Translated from Russian)
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[5]	F. Klein, "Development of mathematics in the 19th century" , Math. Sci. Press (1979) (Translated from German) MR0529278 MR0549187 Zbl 0411.01009
[6]	D.J. Struik, "A concise history of mathematics" , 1–2 , Bell (1954) (Translated from Dutch) MR1834354 MR1109500 MR1156496 MR1052918 MR0919604 MR0736154 MR0753648 MR0610985 MR0207502 MR0103797 MR0093452 MR0093451 MR1527396 MR0026572 Zbl 0715.01004 Zbl 0725.01001 Zbl 0645.01003 Zbl 0528.01001 Zbl 0504.01001 Zbl 0161.00104 Zbl 0136.00206 Zbl 0093.00205 Zbl 0032.09701
[7]	D. Hilbert, "Grundlagen der Geometrie" , Springer (1913) pp. Appendix VIII MR1859422 MR1807508 MR1807507 MR1732507 MR1676305 MR1109913 MR0981143 MR0874532 MR0851072 MR0799771 MR0474006 MR0309913 MR0262046 MR0229120 MR0177322 MR0098003 MR0080308 MR1511181 Zbl 44.0543.02
[8]	, On the foundations of geometry. A collection of classical papers on Lobachevskii geometry , Moscow (1956) (In Russian) (Collection of translations)
[9]	N.V. Efimov, "Höhere Geometrie" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian) MR0120537 Zbl 0211.23702 Zbl 0101.37601 Zbl 0052.37103 Zbl 0036.10002 Zbl 0061.31610
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Comments

Synthetic geometry is closely related to the foundations of geometry.

As is well known, Gauss and J. Bolyai conceived of a non-Euclidean geometry before or at almost the same time as Lobachevskii. The latter, however, was the first to publish his thoughts. For a brief history see, e.g., [a1].

The theory of invariants of groups has been greatly expounded by E. Cartan.

The idea to use the theory of convex bodies, or convex geometry, to solve problems in number theory led Minkowski to the geometry of numbers. It is closely related to the geometric theory of densest packing and thinnest covering (in particular of balls) which traces back to J. Lagrange and Gauss and which plays an eminent role not only in modern geometry but also in algebra and coding theory.

Geometric theories in which discrete aspects are central (e.g. packings and coverings, lattice point problems, etc.) are often subsumed under discrete geometry.

Combinatorial geometry and convex geometry are the main geometric tools in optimization and operations research, which are important branches of modern applied mathematics.

References

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