Invariants

numbers, algebraic expressions, and so forth related to some mathematical object and remaining unchanged under certain transformations of the object or of the frame of reference in which the object is described. In order to characterize a geometric figure and its position in terms of numbers, it is usually necessary to introduce some auxiliary frame of reference or coordinate system. The numbers x₁, x₂, …, x_n obtained in such a system characterize not only the geometric figure under study but also its relation to the coordinate system, and when this system changes, the numbers x₁, x₂, …, x_n are replaced by some other numbers x′₁, x′₂, . . . x′_n. Therefore, if the value of some expression f (x₁, x₂, …, x_n) is characteristic of the figure itself, it should not depend on the coordinate system, that is, the following relation must hold:

f(x₁, x₂, …, x_n) = f(x′₁, x′₂, …, x′_n)

All expressions that satisfy relation (1) are called invariants. For example, the position of a segment M₁M₂ in a plane is defined in a rectangular coordinate system by two pairs of numbers x₁, y₂ and x₂, y₂—the coordinates of its end points M₁ and M₂. If we transform the coordinate system (by translating and rotating its axes), then the points M₁ and M₂ obtain different coordinates x′₁, y′₁ and x′₂, y′₂, but (x₁ - x₂)² + (y₁ - y₂)² = (x′₁ - x′₂)² + (y′₁ - y′₂)². Therefore, the expression (x₁ - x₂)² + (y₁ - y₂)² is an invariant of each transformation of rectangular coordinates. The geometric meaning of this invariant is clear: it is the square of the length of the segment M₁M₂.

A second-order curve in a rectangular coordinate system is given by a second-degree equation

(2) ax² + 2bxy + cy² + 2dx + 2ey + f = 0

whose coefficients may be viewed as numbers that define the curve. Upon transition to another system of rectangular coordinates, these coefficients change, but the expression I = (ac − b²)/(a + c)² retains its value and consequently is an invariant of the curve (2). A similar but more general problem arises in the study of curves and surfaces of higher orders.

The concept of an invariant was used already by the German mathematician L. O. Hesse (1844), but the theory of invariants was developed systematically by the British mathematician J. Sylvester (1851–52), who proposed the term “invariant.” During the second half of the 19th century the theory of invariants was one of the mathematical theories most frequently worked on. In the course of development of this classical theory of invariants, the main efforts of investigators gradually came to be concentrated around the solution of certain “fundamental” problems, the best known of which was stated in the following manner. Consider invariants of a system of forms that are rational integral functions of the coefficients of these forms. Prove that there exists a finite basis for the invariants of each finite system of such forms, that is, a finite system of rational integral numbers in terms of which every other rational integral invariant may be expressed in the form of a rational integral function. The proof for projective invariants was given at the end of the 19th century by the German mathematician D. Hilbert.

An extremely fruitful approach to the concept of invariants results if the n-tuples x₁, x₂, …, x_n and x′₁, x′₂, …, x′_n are considered not as coordinates of a single point with respect to different coordinate systems but rather as the coordinates of different points in the same coordinate system obtained from one another by a motion. The motions of space form a group. Invariants of coordinate transformations are also invariants of the group of motions. Direct generalization leads to the concept of the invariants of any group of transformations. The theory of such invariants is closely related to group theory, especially the theory of group representations.

The concept of the invariants of a group of transformations underlies the well-known systematization of the geometric disciplines according to the groups of transformations whose invariants are studied in these disciplines. For example, the invariants of the group of orthogonal transformations are studied in ordinary Euclidean geometry, the invariants of affine transformations in affine geometry, and the invariants of projective transformations in projective geometry. All one-to-one bicon-tinuous mappings constitute an extremely general group of transformations. The study of the invariants of these so-called topological mappings is the subject of topology. In differential geometry differential invariants are of fundamental importance. Their study led to the creation of the tensor calculus.

In the 20th century the theory of relativity, in which the invariance of physical laws with respect to a certain group of motions is a leading principle, has had a profound influence on the development of the theory of invariants, in particular on the development of the tensor calculus.