geometry as the study of invariants under certain transformations

An approach to geometryMathworldPlanetmath first formulated by Felix Klein in his Erlangen lectures is to describe it as the study of invariantsMathworldPlanetmath under certain allowed transformationsMathworldPlanetmathPlanetmath. This involves taking our space as a set S, and considering a subgroupMathworldPlanetmathPlanetmath G of the group Bij(S), the set of bijectionsMathworldPlanetmath of S. Objects are subsets of S, and we consider two objects A,BS to be equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath if there is an fG such that f(A)=B.

A property P of subsets of S is said to be a geometric property if it is invariant under the action of the group G, which is to say that P(S) is true (or false) if and only if P(g(S)) is true (or false) for every transformation gG. For example, the property of being a straight line is a geometric property in Euclidean geometry. Note that the question whether or not a certin property is geometric depends on the choice of group. For instance, in the case of Euclidean geometry, the property of orthogonality is geometric because, given two lines L1 and L2 and any transformation g which belongs to the Euclidean group, the lines g(L1) and g(L2) are orthogonalMathworldPlanetmathPlanetmathPlanetmath if and only if L1 and L2 are orthogonal. However, if we consider affine geometryMathworldPlanetmath, orthogonality is no longer a geometric property because, given two orthogonal lines L1 and L2, one can find a transformation f which belongs to the affine group such that f(L1) is not orthogonal to f(L2).

Invariants can also be numbers. A real-valued function f whose domain consists of subsets of S is an invariant, or a geometrical quantity if the domain of X is invariant under the action of G and f(X)=f(g(X)) for all subsets X in the domain of f and all transformations gG. Familiar examples from Euclidean geometry are the length of line segmentsMathworldPlanetmath, areas of triangles, and angles. An important feature of the group-theoretic approach to geometry is that one one can use the techniques of invariant theory to systematically find and classify the invariants of a geometrical system. Using this approach, one can start with the description of a geometrical system in terms of a set and a group and rediscover geometric quantities which were originally found by trial and error.

One is not always interested in considering all possible subsets of S. For instance, in algebraic geometryMathworldPlanetmathPlanetmath, one only cares about subsets which can be defined by sytems of algebraic equations. To accommodate this desire, one may revise Klein’s definition by replacing the set S with a suitable categoryMathworldPlanetmath (such as the category of algebraicMathworldPlanetmath subsets) to obtain the definition “geometry is the study of the invariants of a category C under the action of a group G which acts upon this category.” Not only is such an approach popular in contemporary algebraic geometry, it is also useful when discussing such phenomena as duality transforms which map a point in one space to a line in another space and vice-versa. Such a phenomenon is not easily accomodated in a set-theoretic framework, but in terms of category theoryMathworldPlanetmathPlanetmathPlanetmath, the duality transform can be described as a contravariant functorMathworldPlanetmath.

Klein’s definition provides an organizing principle for classifying geometries. Ever since the discovery of non-Euclidean geometry, geometers have been defined and studied many different geometries. Without an organizing principle, the discussion and comparison of these geometries could become confusing. In the next sectionMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, we shall describe several familiar geometric systems from the standpoint of Klein’s definition.

0.1 Basic examples

0.1.1 Euclidean geometry

Euclidean geometry deals with n as a vector spaceMathworldPlanetmath along with a metric d. The allowed transformations are bijections f:nn that preserve the metric, that is, d(𝒙,𝒚)=d(f(𝒙),f(𝒚)) for all 𝒙,𝒚n. Such maps are called isometriesMathworldPlanetmath, and the group is often denoted by Iso(n). Defining a norm by |x|=d(𝒙,𝟎), for 𝒙n, we obtain a notion of length or distanceMathworldPlanetmath. We can also define an inner product 𝒙,𝒚=𝒙𝒚 on n using the standard dot productMathworldPlanetmath (this induces the same norm which can now be defined as |x|=𝒙,𝒙). An inner product leads to a definition of the angle between two vectors 𝒙,𝒚n to be 𝒙𝒚=cos-1(𝒙,𝒚|𝒙||𝒚|). It is clear that since isometries preserve the metric, they preserve distance and angle. As an example, it can be shown that the group Iso(2) consists of translationsMathworldPlanetmathPlanetmath, reflectionsMathworldPlanetmathPlanetmath, glides, and rotations. In general, a member f of Iso(n) has the form f(𝒙)=𝑼𝒙+𝒄, where 𝑼 is an orthogonal n×n matrix and 𝒄n.

0.1.2 Affine geometry

Unlike Euclidean geometry, we are no longer bound to “rigid motion” transformations in affine geometry. Here, we are interested in what happens to geometric objects when they undergo a finite series of “parallel projections”. For example, imagine two Euclidean planesMathworldPlanetmath (2) in 3. Loosely speaking, Euclidean geometry deals with transformations that take objects from one plane to the other, when the planes are parallelMathworldPlanetmathPlanetmathPlanetmath to each other. In affine geometry, the transformation is between two copies of 2, but they are no longer required to be parallel to each other anymore. Objects from one plane will appear to be “stretched” in the other. A circle will turn into an ellipseMathworldPlanetmathPlanetmath, etc…

For 2, in terms of the Kleinian view of geometry, affine geometry consists of the ordinary Euclidean plane, together with a group of transformations that

  1. 1.

    map straight lines to straight lines,

  2. 2.

    map parallel lines to parallel lines, and

  3. 3.

    preserve ratios of lengths of line segments along a given straight line.

Of course, the properties can be generalized to n and n-1 dimensional hyperplanesMathworldPlanetmathPlanetmath. A typical tranformation in an affine geometry is called an affine transformation ( T(𝒙)=𝑨𝒙+𝒃, where xn and 𝑨 is an invertiblePlanetmathPlanetmathPlanetmath n×n real matrix.

0.1.3 Projective geometry

Projective geometryMathworldPlanetmath was motivated by how we see objects in everyday life. For example, parallel train tracks appear to meet at a point far away, even though they are always the same distance apart. In projective geometry, the primary invariant is that of incidence. The notion of parallelism and distance is not present as with Euclidean geometry. There are different ways of approaching projective geometry. One way is to add points of infinityMathworldPlanetmath to Euclidean space. For example, we may form the projective line by adding a point of infinity , called the ideal point, to . We can then create the projective planeMathworldPlanetmath where for each line l2, we attach an ideal point, and two ordinary lines have the same ideal point if and only if they are parallel. The projective plane then consists of the regularPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath plane 2 along with the ideal line, which consists of all ideal points of all ordinary lines. The idea here is to make central projections from a point sending a line to another a bijectiveMathworldPlanetmath map.

Another approach is more algebraic, where we form P(V) where V is a vector space. When V=n, we take the quotient of n+1-{0} where vλv for vn,λ. The allowed transformations is the group PGL(n+1), which is the general linear groupMathworldPlanetmath modulo the subgroup of scalar matrices.

0.1.4 Spherical geometry

Spherical geometry deals with restricting our attention in Euclidean space to the unit sphereMathworldPlanetmath Sn. The role of straight lines is taken by great circlesMathworldPlanetmath. Notions of distance and angles can be easily developed, as well as spherical laws of cosines, the law of sines, and spherical triangles.

Title geometry as the study of invariants under certain transformations
Canonical name GeometryAsTheStudyOfInvariantsUnderCertainTransformations
Date of creation 2013-03-22 18:00:29
Last modified on 2013-03-22 18:00:29
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 6
Author rspuzio (6075)
Entry type Topic
Classification msc 51-01
Classification msc 51-00