Search results
Results from the WOW.Com Content Network
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.
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.
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, the Mazur–Ulam theorem states that if and are normed spaces over R and the mapping: is a surjective isometry, then is affine.It was proved by Stanisław Mazur and Stanisław Ulam in response to a question raised by Stefan Banach.
A Euclidean isometry f of a Euclidean space E defines a linear isometry of the associated vector space (by linear isometry, it is meant an isometry that is also a linear map) in the following way: denoting by Q – P the vector , (if O is an arbitrary point of E, one has
In linear algebra, a branch of mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space. If a norm arises from an inner product then the polarization identity can be used to express this inner product entirely in terms of the norm. The ...
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 mathematics, Liouville's theorem, proved by Joseph Liouville in 1850, [1] is a rigidity theorem about conformal mappings in Euclidean space.It states that every smooth conformal mapping on a domain of R n, where n > 2, can be expressed as a composition of translations, similarities, orthogonal transformations and inversions: they are Möbius transformations (in n dimensions).