|
Quaternion-Kähler Manifold
In differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian 4''n''-manifold whose Riemannian holonomy group is a subgroup of Sp(''n'')·Sp(1) for some n\geq 2. Here Sp(''n'') is the sub-group of SO(4n) consisting of those orthogonal transformations that arise by left-multiplication by some quaternionic n \times n matrix, while the group Sp(1) = S^3 of unit-length quaternions instead acts on quaternionic n-space ^n = ^ by right scalar multiplication. The Lie group Sp(n)\cdot Sp(1) \subset SO(4n) generated by combining these actions is then abstractly isomorphic to p(n) \times Sp(1) _2. Although the above loose version of the definition includes hyperkähler manifolds, the standard convention of excluding these will be followed by also requiring that the scalar curvature be non-zero— as is automatically true if the holonomy group equals the entire group Sp(''n'')·Sp(1). Early history Marcel Berger's 1955 paper on the classif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as classical antiquity, antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Nikolai Lobachevsky, Lobachevsky. The simplest examples of smooth spaces are the Differential geometry of curves, plane and space curves and Differential geometry of surfaces, surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolf Space
The wolf (''Canis lupus''; : wolves), also known as the grey wolf or gray wolf, is a canine native to Eurasia and North America. More than thirty subspecies of ''Canis lupus'' have been recognized, including the dog and dingo, though grey wolves, as popularly understood, only comprise naturally-occurring wild subspecies. The wolf is the largest wild extant member of the family Canidae, and is further distinguished from other ''Canis'' species by its less pointed ears and muzzle, as well as a shorter torso and a longer tail. The wolf is nonetheless related closely enough to smaller ''Canis'' species, such as the coyote and the golden jackal, to produce fertile hybrids with them. The wolf's fur is usually mottled white, brown, grey, and black, although subspecies in the arctic region may be nearly all white. Of all members of the genus ''Canis'', the wolf is most specialized for Pack hunter, cooperative Hunting behavior of gray wolves, game hunting as demonstrated by its p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
