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2. Equivariant mappings from vector product into G-spaces of ϕ-scalars with G = O (n, 1, R)
- Creator:
- Glanc, Barbara, Misiak, Aleksander, and Szmuksta-Zawadzka, Maria
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- G-space, equivariant map, and pseudo-Euclidean geometry
- Language:
- English
- Description:
- There are four kinds of scalars in the n-dimensional pseudo-Euclidean geometry of index one. In this note, we determine all scalars as concomitants of a system of m ≤ n linearly independent contravariant vectors of two so far missing types. The problem is resolved by finding the general solution of the functional equation F(Au 1 , Au 2 , . . . , Au m ) = ϕ (A) · F(u 1 , u 2 , . . . , u m ) using two homomorphisms ϕ from a group G into the group of real numbers R0 = (R \ {0} , ·).
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
3. Equivariant maps between certain G-spaces with G = O(n − 1, 1)
- Creator:
- Misiak, Aleksander and Stasiak, Eugeniusz
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- G-space, equivariant map, vector, scalar, and biscalar
- Language:
- English
- Description:
- In this note, there are determined all biscalars of a system of s ≤ n linearly independent contravariant vectors in n-dimensional pseudo-Euclidean geometry of index one. The problem is resolved by finding a general solution of the functional equation F(Au1 , Au2 , . . . , Aus ) = (sign(det A))F(u1 , u2 ,...,us ) for an arbitrary pseudo-orthogonal matrix A of index one and the given vectors u1 , u2 ,...,us .
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
4. G-space of isotropic directions and G-spaces of ϕ-scalars with G = O(n, 1, ℝ)
- Creator:
- Misiak, Aleksander and Stasiak, Eugeniusz
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- G-space, equivariant map, and pseudo-Euclidean geometry
- Language:
- English
- Description:
- There exist exactly four homomorphisms ϕ from the pseudo-orthogonal group of index one G = O(n, 1, R) into the group of real numbers R0. Thus we have four G-spaces of ϕ-scalars (R, G, hϕ) in the geometry of the group G. The group G operates also on the sphere S n−2 forming a G-space of isotropic directions (S n−2 , G, ∗). In this note, we have solved the functional equation F(A∗q1, A∗q2, . . . , A∗qm) = ϕ(A)·F(q1, q2, . . . , qm) for given independent points q1, q2, . . . , qm ∈ S n−2 with 1 ≤ m ≤ n and an arbitrary matrix A ∈ G considering each of all four homomorphisms. Thereby we have determined all equivariant mappings F : (S n−2 ) m → R.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
5. Invariant metrics on $G$-spaces
- Creator:
- Hajduk, Bogusław and Walczak, Rafał
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- G-space, invariant metric, and slice
- Language:
- English
- Description:
- Let $X$ be a $G$-space such that the orbit space $X/G$ is metrizable. Suppose a family of slices is given at each point of $X$. We study a construction which associates, under some conditions on the family of slices, with any metric on $X/G$ an invariant metric on $X$. We show also that a family of slices with the required properties exists for any action of a countable group on a locally compact and locally connected metric space.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public