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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.
With this distance, the set of isometry classes of -dimensional normed spaces becomes a compact metric space, called the Banach–Mazur compactum. Definitions [ edit ]
It is a linear isometry at the tangent space of every point on (), that is, it is an isometry on the infinitesimal patches. It preserves the curvature tensor at the tangent space of every point on B r ( x ) {\displaystyle B_{r}(x)} , that is, it preserves how the infinitesimal patches fit together.
Write I(S) for the set of integral linear combinations of S, and I 0 (S) for the subset of degree 0 elements of I(S). Suppose that τ is an isometry from I 0 (S) to the degree 0 virtual characters of G. Then τ is called coherent if it can be extended to an isometry from I(S) to characters of G and I 0 (S) is non-zero.
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.
A discrete isometry group is an isometry group such that for every point of the space the set of images of the point under the isometries is a discrete set. In pseudo-Euclidean space the metric is replaced with an isotropic quadratic form ; transformations preserving this form are sometimes called "isometries", and the collection of them is ...
In mathematical functional analysis a partial isometry is a linear map between Hilbert spaces such that it is an isometry on the orthogonal complement of its kernel. The orthogonal complement of its kernel is called the initial subspace and its range is called the final subspace. Partial isometries appear in the polar decomposition.
A partial linear space is an incidence structure for which the following axioms are true: [3] Every pair of distinct points determines at most one line. Every line contains at least two distinct points. In a partial linear space it is also true that every pair of distinct lines meet in at most one point.