The growth of the first non-Euclidean filling volume function of the quaternionic Heisenberg group

2019 ◽  
Vol 19 (3) ◽  
pp. 415-420
Author(s):  
Moritz Gruber
Keyword(s):  
Growth Rate ◽  
Euclidean Space ◽  
Heisenberg Group ◽  
Filling Volume ◽  
Volume Function ◽  
Quaternionic Heisenberg Group ◽  
Volume Functions

Abstract The filling volume functions of the n-th quaternionic Heisenberg group grow, up to dimension n, as fast as the ones of the Euclidean space. We identify the growth rate of the filling volume function in dimension n + 1, which is strictly faster than the growth rate of the (n + 1)-dimensional filling volume function of the Euclidean space.

scholarly journals CONJUGATE POINTS ON THE QUATERNIONIC HEISENBERG GROUP

2003 ◽  
Vol 40 (1) ◽  
pp. 61-72 ◽  
Author(s):  
Chang-Rim Jang ◽  
Jun-Kon Kim ◽  
Yeon-Wook Kim ◽  
Keun Park
Keyword(s):  
Heisenberg Group ◽  
Conjugate Points ◽  
Quaternionic Heisenberg Group

scholarly journals TSALLIS ENTROPY COMPOSITION AND THE HEISENBERG GROUP

2013 ◽  
Vol 10 (07) ◽  
pp. 1350032 ◽  
Author(s):  
NIKOS KALOGEROPOULOS
Keyword(s):  
Phase Space ◽  
Euclidean Space ◽  
Heisenberg Group ◽  
Polynomial Growth ◽  
Three Dimensional ◽  
Tsallis Entropy ◽  
Geometric Framework ◽  
Fractal Properties ◽  
Composition Property ◽  
Riemannian Spaces

We present an embedding of the Tsallis entropy into the three-dimensional Heisenberg group, in order to understand the meaning of generalized independence as encoded in the Tsallis entropy composition property. We infer that the Tsallis entropy composition induces fractal properties on the underlying Euclidean space. Using a theorem of Milnor/Wolf/Tits/Gromov, we justify why the underlying configuration/phase space of systems described by the Tsallis entropy has polynomial growth for both discrete and Riemannian cases. We provide a geometric framework that elucidates Abe's formula for the Tsallis entropy, in terms the Pansu derivative of a map between sub-Riemannian spaces.

scholarly journals Non-Umbilical Quaternionic Contact Hypersurfaces in Hyper-Kähler Manifolds

2017 ◽  
Vol 2019 (18) ◽  
pp. 5649-5673
Author(s):  
Stefan Ivanov ◽  
Ivan Minchev ◽  
Dimiter Vassilev
Keyword(s):  
Heisenberg Group ◽  
Kähler Manifold ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Totally Umbilical ◽  
Kahler Manifold ◽  
Compact Case ◽  
Quaternionic Heisenberg Group

Abstract It is shown that any compact quaternionic contact (qc) hypersurfaces in a hyper-Kähler manifold which is not totally umbilical has an induced qc structure, locally qc homothetic to the standard 3-Sasakian sphere. In the non-compact case, it is proved that a seven-dimensional everywhere non-umbilical qc-hypersurface embedded in a hyper-Kähler manifold is qc-conformal to a qc-Einstein structure which is locally qc-equivalent to the 3-Sasakian sphere, the quaternionic Heisenberg group or the hyperboloid.

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