The Riemannian metric on the manifold of positive definite matrices is defined by a kernel function φ in the form K D(H,K) = ∑ i,j φ(λi, λj) −1TrPiHPjK when ∑ i λiPi is the spectral decomposition of the foot point D and the Hermitian matrices H,K are tangent vectors. For such kernel metrics the tangent space has an orthogonal decomposition. The pull-back of a kernel metric under a mapping D 7→ ...

‎In this paper we try to extend geometric concepts in the context of operator valued tensors‎. ‎To this end‎, ‎we aim to replace the field of scalars $ mathbb{R} $ by self-adjoint elements of a commutative $ C^star $-algebra‎, ‎and reach an appropriate generalization of geometrical concepts on manifolds‎. ‎First‎, ‎we put forward the concept of operator-valued tensors and extend semi-Riemannian...

The Riemannian metric on the manifold of positive de nite matrices is de ned by a kernel function in the formK D(H;K) = P i;j ( i; j) TrPiHPjK when P i iPi is the spectral decomposition of the foot point D and the Hermitian matrices H;K are tangent vectors. For such kernel metrics the tangent space has an orthogonal decomposition. The pull-back of a kernel metric under a mapping D 7! G(D) is a ...

The paper interprets the concept “operator in separable complex Hilbert space” (particalry, “Hermitian operator” as “quantity” is defined “classical” quantum mechanics) by that of “quantum information”. As far wave function characteristic probability (density) distribution for all possible values a certain quantity to be measured, definition mechanics means any unitary change distribution. It c...

We study some Weingarten spacelike hypersurfaces in a de Sitter space S 1 (1). If the Weingarten spacelike hypersurfaces have two distinct principal curvatures, we obtain two classification theorems which give some characterization of the Riemannian product H(1−coth ̺)× S(1 − tanh ̺), 1 < k < n − 1 in S 1 (1), the hyperbolic cylinder H(1 − coth ̺) × S(1 − tanh ̺) or spherical cylinder H(1 − coth ̺)×...

We consider the sub-Riemannian metric g h on S 3 provided by the restriction of the Riemannian metric of curvature 1 to the plane distribution orthogonal to the Hopf vector field. We compute the geodesics associated to the Carnot-Carathéodory distance and we show that, depending on their curvature, they are closed or dense subsets of a Clifford torus. We study area-stationary surfaces with or w...

The Riemannian metric on the manifold of positive definite matrices is defined by a kernel function φ in the form K D(H,K) = ∑ i,j φ(λi, λj) −1TrPiHPjK when ∑ i λiPi is the spectral decomposition of the foot point D and the Hermitian matrices H,K are tangent vectors. For such kernel metrics the tangent space has an orthogonal decomposition. The pull-back of a kernel metric under a mapping D 7→ ...

Geodesic orbit spaces (or g.o. spaces) are defined as those homogeneous Riemannian (M=G∕H,g) whose geodesics orbits of one-parameter subgroups G. The corresponding metric g is called a geodesic metric. We study the form (Sp(n)∕Sp(n1)×⋯×Sp(ns),g), with 0<n1+⋯+ns≤n. Such include spheres, quaternionic Stiefel manifolds, Grassmann manifolds and flag manifolds. present work contribution to (G∕H,g) H...

We consider the unit tangent sphere bundle of Riemannian manifold ( M, g ) with g-natural metric G̃ and we equip it to an almost contact B-metric structure. Considering this structure, we show that there is a direct correlation between the Riemannian curvature tensor of ( M, g ) and local symmetry property of G̃. More precisely, we prove that the flatness of metric g is necessary and sufficien...

Abstract In this paper, we prove the Lipschitz regularity of continuous harmonic maps from a finite-dimensional Alexandrov space to compact smooth Riemannian manifold. This solves conjecture F. H. Lin in [38]. The proof extends argument Huang-Wang [28].