Statistical structures on metric path spaces

Crâşmăreanu, Mircea; Hreƫcanu, Cristina E.

doi:10.1007/s11401-012-0745-9

Cited by 4 publications

(4 citation statements)

References 11 publications

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“…• Statistical manifolds, where the tensor∇g is symmetric in all its entries and connection∇ is torsion-free, e.g., [9,16,17]. These conditions are equivalent to T ∧ = T and T * = T. The theory of affine hypersurfaces in R n+1 is a natural source of such manifolds; they also find applications in theory of probability and statistics.…”

Section: Introductionmentioning

confidence: 99%

“…is an analog of (2), where S is replaced by the mixed scalar curvature S mix , see (9), for the affine connection∇ = ∇ + T. The physical meaning of (4) is discussed in [1] for the case of T = 0. In view of the formula S = S mix + S ⊤ + S ⊥ , where S ⊤ and S ⊥ are the scalar curvatures along the distributions D and D, one can combine the actions (2) and 4 (S + ε S mix − 2 Λ) + L d vol g with ε ∈ R, whose critical points may describe geometry of the space-time in an extended theory of gravity.…”

Section: Introductionmentioning

confidence: 99%

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The Einstein-Hilbert Type Action on Pseudo-Riemannian Almost-Product Manifolds

Rovenski¹,

Zawadzki²

2019

Z. mat. fiz. anal. geom.

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We develop variation formulas on almost-product (e.g. foliated) pseudo-Riemannian manifolds, and we consider variations of metric preserving orthogonality of the distributions. These formulae are applied to Einstein-Hilbert type actions: the total mixed scalar curvature and the total extrinsic scalar curvature of a distribution. The obtained Euler-Lagrange equations admit a number of solutions, e.g., twisted products, conformal submersions and isoparametric foliations. The paper generalizes recent results about the actions on codimension-one foliations for the case of arbitrary (co)dimension.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Einstein-Hilbert Type Action on Pseudo-Riemannian Almost-Product Manifolds

Rovenski¹,

Zawadzki²

2019

Z. mat. fiz. anal. geom.

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show abstract

“…• Statistical manifolds, where the tensor∇g is symmetric in all its entries and connection∇ is torsion-free, e.g., [10,18,19]. These conditions are equivalent to T ∧ = T and T * = T. The theory of affine hypersurfaces in R n+1 is a natural source of such manifolds; they also find applications in theory of probability and statistics.…”

Section: State Of the Artmentioning

confidence: 99%

“…If a distribution is spanned by a unit vector field N , i.e., N, N = N ∈ {−1, 1}, then S mix = N Ric N,N , where Ric N,N is the Ricci curvature in the N -direction. If dim M = 2 and dim D = 1, then obviously 2S mix = S. If T = 0 then S mix reduces to the mixed scalar curvature S mix of ∇, see (10), which can be defined as a sum of sectional curvatures of planes that non-trivially intersect with both of the distributions. Investigation of S mix led to multiple results regarding the existence of foliations and submersions with interesting geometry, e.g., integral formulas and splitting results, curvature prescribing and variational problems, see survey [23].…”

Section: Objectivesmentioning

confidence: 99%

The Mixed Scalar Curvature of Almost-Product Metric-Affine Manifolds, II

Rovenski

Zawadzki

2021

Results Math

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We continue our study of the mixed Einstein–Hilbert action as a functional of a pseudo-Riemannian metric and a linear connection. Its geometrical part is the total mixed scalar curvature on a smooth manifold endowed with a distribution or a foliation. We develop variational formulas for quantities of extrinsic geometry of a distribution on a metric-affine space and use them to derive Euler–Lagrange equations (which in the case of space-time are analogous to those in Einstein–Cartan theory) and to characterize critical points of this action on vacuum space-time. Together with arbitrary variations of metric and connection, we consider also variations that partially preserve the metric, e.g., along the distribution, and also variations among distinguished classes of connections (e.g., statistical and metric compatible, and this is expressed in terms of restrictions on contorsion tensor). One of Euler–Lagrange equations of the mixed Einstein–Hilbert action is an analog of the Cartan spin connection equation, and the other can be presented in the form similar to the Einstein equation, with Ricci curvature replaced by the new Ricci type tensor. This tensor generally has a complicated form, but is given in the paper explicitly for variations among semi-symmetric connections.

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The Mixed Scalar Curvature of Almost-Product Metric-Affine Manifolds

Rovenski

Zawadzki

2018

Results Math

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A pseudo-Riemannian manifold endowed with k > 2 orthogonal complementary distributions (called a Riemannian almost multi-product structure) appears in such topics as multiply warped products, the webs composed of several foliations, Dupin hypersurfaces and in studies of the curvature and Einstein equations. In this article, we consider the following two problems on the mixed scalar curvature of a Riemannian almost multi-product manifold with a linear connection: a) integral formulas and applications to splitting of manifolds, b) variation formulas and applications to the mixed Einstein-Hilbert action, and we generalize certain results on the mixed scalar curvature of pseudo-Riemannian almost product manifolds.

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Statistical structures on metric path spaces

Cited by 4 publications

References 11 publications

The Einstein-Hilbert Type Action on Pseudo-Riemannian Almost-Product Manifolds

The Einstein-Hilbert Type Action on Pseudo-Riemannian Almost-Product Manifolds

The Mixed Scalar Curvature of Almost-Product Metric-Affine Manifolds, II

The Mixed Scalar Curvature of Almost-Product Metric-Affine Manifolds

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