这个内积就是所谓的Riemannian metric。而由这样一个metric定义的manifold则被称为Riemannian manifold。 一个栗子,在 [2] 中定义了一个 Fisher-Rao norm用于度量神经网络的complexity。 参数θ 的Fisher-Rao norm即 ‖θ‖fr2=⟨θ,I(θ)θ⟩ ,这里的 I(θ)=E[∇θℓ(fθ(X),Y)ℓ(fθ(X),Y...
设g 是流形 M 上的黎曼度量(Riemannian metric)。假设在 M 上的局部坐标 {ui} 下,度量具有局部分量( local components) gij=g(∂∂ui,∂∂uj) ,则我们将克里斯托费尔符号(Christoffel symbol)定义为: Γijk:=12gkl(∂gjl∂ui+∂gil∂uj−∂gij∂ul) 其中是矩阵的第gkl是矩阵[g]−...
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RegisterLog in Sign up with one click: Facebook Twitter Google Share on Facebook Riemannian manifold (redirected fromRiemannian manifolds) [rē′män·ē·ən ′man·ə‚fōld] (mathematics) A differentiable manifold where the tangent vectors about each point have an inner product so defined...
The following sections are included:OutlineDefinition of the Frame Bundle for a SurfaceLocal Homogeneous FieldsSurfaces of Zero CurvatureSurfaces of Constant Negative CurvatureRemarks on CurvatureIndefinite Two-Dimensional MetricsThe Field ConceptThe de Sitter Metric (K< 0)Geodesics of the de Sitter ...
e., to the question: to what extent is a tensor field on a closed Riemannian manifold determined by its integrals over all closed geodesies? Anosov manifolds, i. e., closed Riemannian manifolds with geodesic flow of Anosov type constitute the most natural class for investigating the latter ...
Thus, in the eyes of the metric, the unit basis vectors ei = /i are orthogonal; that is, ei, ej = ±ij. Note The non-degeneracy condition in Definition 6.1 is equivalent to the requirement that the locally defined quantities g = det(gij) are nowhere zero. ...
Here we explore an extension of Riemannian geometry using a complex Hermitian metric tensor. We find that the standard electromagnetic field naturally appears along with two additional fields, which act as mass and charge sources. A first paper set up the basic geometry and derived the Christoffel...
Such result can be obtained by noting (i)Df(x)[η]=⟨PTxM∇f(x),η⟩F=⟨gradf(x),η⟩xfor [15, (B.2)], and (ii) there exists a constantα>0such that‖η‖F≤α‖η‖xfor allx∈Mby smoothness of the Riemannian metric and compactness ofM. ...
1) Riemannian metric 黎曼度量 1. TheRiemannian metricof perspective in stereo graphic projection is introduced and a property of the continual cut vector field on the spherical surface is discussed. 介绍了在球极投影下引入的黎曼度量,并介绍了这种度量的一些性质,指出了这种度量与从平直的欧几里德空间标...