scholarly journals Chern connection of a pseudo-Finsler metric as a family of affine connections

2014 ◽  
Vol 84 (1-2) ◽  
pp. 29-43 ◽  
Author(s):  
MIGUEL ANGEL JAVALOYES
Keyword(s):  
Finsler Metric ◽  
Chern Connection ◽  
Affine Connections

scholarly journals Anisotropic tensor calculus

2019 ◽  
Vol 16 (supp02) ◽  
pp. 1941001 ◽  
Author(s):  
Miguel Angel Javaloyes
Keyword(s):  
Curvature Tensor ◽  
Finsler Metric ◽  
Lie Derivative ◽  
Classical Concept ◽  
Tensor Calculus ◽  
Chern Connection ◽  
Linear Connections ◽  
Anisotropic Curvature ◽  
Levi Civita Connection ◽  
Anisotropic Tensor

We introduce the anisotropic tensor calculus, which is a way of handling tensors that depends on the direction remaining always in the same class. This means that the derivative of an anisotropic tensor is a tensor of the same type. As an application we show how to define derivations using anisotropic linear connections in a manifold. In particular, we show that the Chern connection of a Finsler metric can be interpreted as the Levi-Civita connection and we introduce the anisotropic curvature tensor. We also relate the concept of anisotropic connection with the classical concept of linear connections in the vertical bundle. Furthermore, we also introduce the concept of anisotropic Lie derivative.

scholarly journals Color algebras and affine connections on S6

1992 ◽  
Vol 149 (1) ◽  
pp. 234-261 ◽  
Author(s):  
Alberto Elduque ◽  
Hyo Chul Myung
Keyword(s):  
Affine Connections

A geometric approach to quantum circuit lower bounds

2006 ◽  
Vol 6 (3) ◽  
pp. 213-262 ◽  
Author(s):  
M.A. Nielsen
Keyword(s):  
Lower Bounds ◽  
Geometric Approach ◽  
Quantum Circuit ◽  
Geodesic Equation ◽  
Initial Position ◽  
Minimal Length ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Minimal Size ◽  
Circuit Lower Bounds

What is the minimal size quantum circuit required to exactly implement a specified n-qubit unitary operation, U, without the use of ancilla qubits? We show that a lower bound on the minimal size is provided by the length of the minimal geodesic between U and the identity, I, where length is defined by a suitable Finsler metric on the manifold SU(2^n). The geodesic curves on these manifolds have the striking property that once an initial position and velocity are set, the remainder of the geodesic is completely determined by a second order differential equation known as the geodesic equation. This is in contrast with the usual case in circuit design, either classical or quantum, where being given part of an optimal circuit does not obviously assist in the design of the rest of the circuit. Geodesic analysis thus offers a potentially powerful approach to the problem of proving quantum circuit lower bounds. In this paper we construct several Finsler metrics whose minimal length geodesics provide lower bounds on quantum circuit size. For each Finsler metric we give a procedure to compute the corresponding geodesic equation. We also construct a large class of solutions to the geodesic equation, which we call \emph{Pauli geodesics}, since they arise from isometries generated by the Pauli group. For any unitary U diagonal in the computational basis, we show that: (a) provided the minimal length geodesic is unique, it must be a Pauli geodesic; (b) finding the length of the minimal Pauli geodesic passing from I to U is equivalent to solving an exponential size instance of the closest vector in a lattice problem (CVP); and (c) all but a doubly exponentially small fraction of such unitaries have minimal Pauli geodesics of exponential length.

scholarly journals Hooke’s law in statistical manifolds and divergences

2000 ◽  
Vol 159 ◽  
pp. 1-24 ◽  
Author(s):  
Masayuki Henmi ◽  
Ryoichi Kobayashi
Keyword(s):  
Classical Mechanics ◽  
Riemannian Metric ◽  
Legendre Transform ◽  
Point Function ◽  
Hooke’S Law ◽  
Affine Connections ◽  
Statistical Manifolds ◽  
Dual Affine ◽  
Hooke's Law

The concept of the canonical divergence is defined for dually flat statistical manifolds in terms of the Legendre transform between dual affine coordinates. In this article, we introduce a new two point function defined for any triple (g,∇, ∇*) of a Riemannian metric g and two affine connections ∇ and ∇*. We show that this interprets the canonical divergence without refering to the existence of special coordinates (dual affine coordinates) but in terms of only classical mechanics concerning ∇- and ∇*-geodesics. We also discuss the properties of the two point function and show that this shares some important properties with the canonical divergence defined on dually flat statistical manifolds.

ON THE GAUSS–BONNET FORMULA IN RIEMANN–FINSLER GEOMETRY

2002 ◽  
Vol 34 (3) ◽  
pp. 329-340 ◽  
Author(s):  
BRAD LACKEY
Keyword(s):  
Finsler Space ◽  
Finsler Geometry ◽  
Euler Class ◽  
Constant Volume ◽  
Finsler Spaces ◽  
Chern Connection ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Riemannian Case ◽  
Torsion Free

Using Chern's method of transgression, the Euler class of a compact orientable Riemann–Finsler space is represented by polynomials in the connection and curvature matrices of a torsion-free connection. When using the Chern connection (and hence the Christoffel–Levi–Civita connection in the Riemannian case), this result extends the Gauss–Bonnet formula of Bao and Chern to Finsler spaces whose indicatrices need not have constant volume.

On Affine Connections Induced on the (1, 1)-Tensor Bundle

2018 ◽  
Vol 39 (4) ◽  
pp. 683-694
Author(s):  
Murat Altunbas ◽  
Aydin Gezer
Keyword(s):  
Affine Connections ◽  
Tensor Bundle
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