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A path isometry or arcwise isometry is a map which preserves the lengths of curves; such a map is not necessarily an isometry in the distance preserving sense, and it need not necessarily be bijective, or even injective. [5] [6] This term is often abridged to simply isometry, so one should take care to determine from context which type is intended.
If and are two finite-dimensional normed spaces with the same dimension, let (,) denote the collection of all linear isomorphisms :. Denote by ‖ ‖ the operator norm of such a linear map — the maximum factor by which it "lengthens" vectors.
Two theorems in the mathematical field of Riemannian geometry bear the name Myers–Steenrod theorem, both from a 1939 paper by Myers and Steenrod.The first states that every distance-preserving surjective map (that is, an isometry of metric spaces) between two connected Riemannian manifolds is a smooth isometry of Riemannian manifolds.
Cartan proved the local version. Ambrose proved a global version that allows for isometries between general Riemannian manifolds with varying curvature, in 1956. [2] This was further generalized by Hicks to general manifolds with affine connections in their tangent bundles, in 1959. [3] A statement and proof of the theorem can be found in [4]
In mathematics, a unitary transformation is a linear isomorphism that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation.
In other words, an isometry is an injective partial isometry. Any finite-dimensional partial isometry can be represented, in some choice of basis, as a matrix of the form A = ( V 0 ) {\displaystyle A={\begin{pmatrix}V&0\end{pmatrix}}} , that is, as a matrix whose first rank ( A ) {\displaystyle \operatorname {rank} (A)} columns form an ...
An isometry V is said to be pure if, in the notation of the above proof, = {}. The multiplicity of a pure isometry V is the dimension of the kernel of V*, i.e. the cardinality of the index set A in the Wold decomposition of V. In other words, a pure isometry of multiplicity N takes the form
For example, for the general linear group GL, a maximal torus is the subgroup D of invertible diagonal matrices, whose normalizer is the generalized permutation matrices (matrices in the form of permutation matrices, but with any non-zero numbers in place of the '1's), and whose Weyl group is the symmetric group.