Invariant (mathematics)
In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects.[1][2] The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class.[3]
For other uses, see Invariant (disambiguation).Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an important step in the process of classifying mathematical objects.[2][3]
The integral of the Gaussian curvature of a two-dimensional is invariant under changes of the Riemannian metric . This is the Gauss–Bonnet theorem.
Riemannian manifold
Invariant set[edit]
A subset S of the domain U of a mapping T: U → U is an invariant set under the mapping when Note that the elements of S are not fixed, even though the set S is fixed in the power set of U. (Some authors use the terminology setwise invariant,[8] vs. pointwise invariant,[9] to distinguish between these cases.)
For example, a circle is an invariant subset of the plane under a rotation about the circle's center. Further, a conical surface is invariant as a set under a homothety of space.
An invariant set of an operation T is also said to be stable under T. For example, the normal subgroups that are so important in group theory are those subgroups that are stable under the inner automorphisms of the ambient group.[10][11][12]
In linear algebra, if a linear transformation T has an eigenvector v, then the line through 0 and v is an invariant set under T, in which case the eigenvectors span an invariant subspace which is stable under T.
When T is a screw displacement, the screw axis is an invariant line, though if the pitch is non-zero, T has no fixed points.
In probability theory and ergodic theory, invariant sets are usually defined via the stronger property [13][14][15] When the map is measurable, invariant sets form a sigma-algebra, the invariant sigma-algebra.
