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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.
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]
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.
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.
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
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 ...
Cartan connection. Cartan-Hadamard space is a complete, simply-connected, non-positively curved Riemannian manifold.. Cartan–Hadamard theorem is the statement that a connected, simply connected complete Riemannian manifold with non-positive sectional curvature is diffeomorphic to R n via the exponential map; for metric spaces, the statement that a connected, simply connected complete ...
In linear algebra, the restricted isometry property (RIP) characterizes matrices which are nearly orthonormal, at least when operating on sparse vectors. The concept was introduced by Emmanuel Candès and Terence Tao [ 1 ] and is used to prove many theorems in the field of compressed sensing . [ 2 ]