Inverse problem of left invariant sprays on Lie groups

2021 ◽  
pp. 2150076
Author(s):  
Libing Huang ◽  
Xiaohuan Mo
Keyword(s):  
Inverse Problem ◽  
Lie Groups ◽  
Lie Group ◽  
Positive Definite ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Riemann Curvature

In this paper, we discuss inverse problem in spray geometry. We find infinitely many sprays with non-diagonalizable Riemann curvature on a Lie group, these sprays are not induced by Finsler metrics. We also study left invariant sprays with non-vanishing spray vectors on Lie groups. We prove that if such a spray [Formula: see text] on a Lie group [Formula: see text] satisfies that [Formula: see text] is commutative or [Formula: see text] is projective, then [Formula: see text] is not induced by any (not necessary positive definite) left invariant Finsler metric. Finally, we construct an abundance of the left invariant sprays on Lie groups which satisfy the conditions in above result.

Inverse problem of sprays with scalar curvature

2019 ◽  
Vol 30 (09) ◽  
pp. 1950041
Author(s):  
Ying Li ◽  
Xiaohuan Mo ◽  
Yaoyong Yu
Keyword(s):  
Inverse Problem ◽  
Scalar Curvature ◽  
Finsler Geometry ◽  
Positive Definite ◽  
Finsler Metric ◽  
Open Domain ◽  
Isotropic Curvature ◽  
Differential Manifold

Every Finsler metric on a differential manifold induces a spray. The converse is not true. Therefore, it is one of the most fundamental problems in spray geometry to determine whether a spray is induced by a Finsler metric which is regular, but not necessary positive definite. This problem is called inverse problem. This paper discuss inverse problem of sprays with scalar curvature. In particular, we show that if such a spray [Formula: see text] on a manifold is of vanishing [Formula: see text]-curvature, but [Formula: see text] has not isotropic curvature, then [Formula: see text] is not induced by any (not necessary positive definite) Finsler metric. We also find infinitely many sprays on an open domain [Formula: see text] with scalar curvature and vanishing [Formula: see text]-curvature, but these sprays have no isotropic curvature. This contrasts sharply with the situation in Finsler geometry.

Sprays of isotropic curvature

2018 ◽  
Vol 29 (01) ◽  
pp. 1850003 ◽  
Author(s):  
Benling Li ◽  
Zhongmin Shen
Keyword(s):  
Inverse Problem ◽  
Scalar Curvature ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Geometric Quantity ◽  
Zero Curvature ◽  
Isotropic Curvature

In this paper, a new notion of isotropic curvature for sprays is introduced. We show that for a spray of scalar curvature, it is of isotropic curvature if and only if the non-Riemannian quantity [Formula: see text] vanishes. In fact, it is the first geometric quantity to show the spray of isotropic curvature even in the Finslerian case. How to determine a spray is induced by a Finsler metric or not is an interesting inverse problem. We study this problem when the spray is of isotropic curvature and show that a spray of zero curvature can be induced by a group of Finsler metrics. Further, an efficient way is given to construct a family of sprays of isotropic curvature which cannot be induced by any Finsler metric.

scholarly journals Some groups with T1 primitive ideal spaces

1985 ◽  
Vol 38 (1) ◽  
pp. 55-64 ◽  
Author(s):  
A. L. Carey ◽  
W. Moran
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Solvable Lie Groups ◽  
Locally Compact ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

scholarly journals The Higher Order Riesz Transform andBMOType Space Associated with Schrödinger Operators on Stratified Lie Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Yu Liu ◽  
Jianfeng Dong
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Riesz Transform ◽  
Integral Operators ◽  
Higher Order ◽  
Reverse Hölder Class ◽  
Stratified Lie Group ◽  
Homogeneous Dimension ◽  
Stratified Lie Groups ◽  
Reverse Hölder

Assume thatGis a stratified Lie group andQis the homogeneous dimension ofG. Let-Δbe the sub-Laplacian onGandW≢0a nonnegative potential belonging to certain reverse Hölder classBsfors≥Q/2. LetL=-Δ+Wbe a Schrödinger operator on the stratified Lie groupG. In this paper, we prove the boundedness of some integral operators related toL, such asL-1∇2,L-1W, andL-1(-Δ) on the spaceBMOL(G).

ON THE MODEL THEORY OF THE LOGARITHMIC FUNCTION IN COMPACT LIE GROUPS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350055
Author(s):  
SONIA L'INNOCENTE ◽  
FRANÇOISE POINT ◽  
CARLO TOFFALORI
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Model Theory ◽  
Effective Model ◽  
Logarithmic Function ◽  
Compact Lie Groups ◽  
Model Completeness ◽  
Schanuel's Conjecture ◽  
Logarithm Function ◽  
Natural Expansion

Given a compact linear Lie group G, we form a natural expansion of the theory of the reals where G and the graph of a logarithm function on G live. We prove its effective model-completeness and decidability modulo a suitable variant of Schanuel's Conjecture.

THE FERMI–WALKER DERIVATIVE IN LIE GROUPS

2013 ◽  
Vol 10 (07) ◽  
pp. 1320011 ◽  
Author(s):  
FATMA KARAKUŞ ◽  
YUSUF YAYLI
Keyword(s):  
Vector Field ◽  
Lie Groups ◽  
Lie Group ◽  
Necessary Conditions ◽  
Rotating Frame ◽  
Darboux Vector

In this study, Fermi–Walker derivative, Fermi–Walker parallelism, non-rotating frame, Fermi–Walker termed Darboux vector concepts are given for Lie groups in E4. First, we get any Frénet curve and any vector field along the Frénet curve in a Lie group. Then, the Fermi–Walker derivative is defined for the Lie group. Fermi–Walker derivative and Fermi–Walker parallelism are analyzed in Lie groups. Finally, the necessary conditions for Fermi–Walker parallelism are explained.

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.

CRITERION OF PROPER ACTIONS FOR 3-STEP NILPOTENT LIE GROUPS

2007 ◽  
Vol 18 (07) ◽  
pp. 783-795 ◽  
Author(s):  
TARO YOSHINO
Keyword(s):  
Homogeneous Space ◽  
Lie Groups ◽  
Lie Group ◽  
Closed Subgroup ◽  
Nilpotent Lie Groups ◽  
Nilpotent Lie Group ◽  
Proper Actions ◽  
Affirmative Solution

For a nilpotent Lie group G and its closed subgroup L, Lipsman [13] conjectured that the L-action on some homogeneous space of G is proper in the sense of Palais if and only if the action is free. Nasrin [14] proved this conjecture assuming that G is a 2-step nilpotent Lie group. However, Lipsman's conjecture fails for the 4-step nilpotent case. This paper gives an affirmative solution to Lipsman's conjecture for the 3-step nilpotent case.

Parabolic Higgs Bundles for Real Reductive Lie Groups

2018 ◽  
pp. 653-680 ◽  
Author(s):  
Ignasi Mundet i Riera
Keyword(s):  
Riemann Surface ◽  
Lie Groups ◽  
Lie Group ◽  
Vector Bundles ◽  
Structure Group ◽  
Higgs Bundles ◽  
Local Systems ◽  
Parabolic Structure ◽  
Reductive Lie Groups ◽  
Reductive Lie Group

This chapter explains the correspondence between local systems on a punctured Riemann surface with the structure group being a real reductive Lie group G, and parabolic G-Higgs bundles. The chapter describes the objects involved in this correspondence, taking some time to motivate them by recalling the definitions of G-Higgs bundles without parabolic structure and of parabolic vector bundles. Finally, it explains the relevant polystability condition and the correspondence between local systems and Higgs bundles.

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