Cut locus of a left invariant Riemannian metric on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si6.gif" display="inline" overflow="scroll"><mml:msub><mml:mrow><mml:mstyle mathvariant="normal"><mml:mi>SO</mml:mi></mml:mstyle></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:math> in the axisymmetric case

Podobryaev, A. V.; Sachkov, Yu. L.

doi:10.1016/j.geomphys.2016.09.005

Cited by 20 publications

(22 citation statements)

References 7 publications

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“…This coincides with results in the literature (see e.g. [1,22]). The conjugate time (for a general geodesic, not necessarily a relative equilibrium) is given in e.g.…”

Section: A Rotation Matrix Insupporting

confidence: 93%

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Conjugate points for systems of second-order ordinary differential equations

Hajdu

Mestdag

2019

Int. J. Geom. Methods Mod. Phys.

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We recall the notion of Jacobi fields, as it was extended to systems of second-order ordinary differential equations. Two points along a base integral curve are conjugate if there exists a non-trivial Jacobi field along that curve that vanishes on both points. Based on arguments that involve the eigendistributions of the Jacobi endomorphism, we discuss conjugate points for a certain generalization (to the current setting) of locally symmetric spaces. Next, we study conjugate points along relative equilibria of Lagrangian systems with a symmetry Lie group. We end the paper with some examples and applications.

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“…This coincides with results in the literature (see e.g. [1,22]). The conjugate time (for a general geodesic, not necessarily a relative equilibrium) is given in e.g.…”

Section: A Rotation Matrix Insupporting

confidence: 93%

“…The conjugate time (for a general geodesic, not necessarily a relative equilibrium) is given in e.g. Theorem 1 of [22]. In their notations, our current situation (0, 0, Ω) ∈ g coincides with the conjugate momentum p = (0, 0, I 3 Ω) ∈ g * .…”

Section: A Rotation Matrix Inmentioning

confidence: 99%

Conjugate points for systems of second-order ordinary differential equations

Hajdu

Mestdag

2019

Int. J. Geom. Methods Mod. Phys.

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“…The geodesics can be described explicitly as follows [58]: the geodesic starting from the identity with tangent vector v is given by…”

Section: Complexity Of One Qubitmentioning

confidence: 99%

“…For the special case of the vacuum state of the Hamiltonian (57) we have a = λ and b = 0. Explicitly evaluating the covariance matrix for the wavefunction (58) we obtain…”

Section: Single Harmonic Oscillatormentioning

confidence: 99%

Quantum computational complexity from quantum information to black holes and back

Chapman

Policastro

2022

Eur. Phys. J. C

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Quantum computational complexity estimates the difficulty of constructing quantum states from elementary operations, a problem of prime importance for quantum computation. Surprisingly, this quantity can also serve to study a completely different physical problem – that of information processing inside black holes. Quantum computational complexity was suggested as a new entry in the holographic dictionary, which extends the connection between geometry and information and resolves the puzzle of why black hole interiors keep growing for a very long time. In this pedagogical review, we present the geometric approach to complexity advocated by Nielsen and show how it can be used to define complexity for generic quantum systems; in particular, we focus on Gaussian states in QFT, both pure and mixed, and on certain classes of CFT states. We then present the conjectured relation to gravitational quantities within the holographic correspondence and discuss several examples in which different versions of the conjectures have been tested. We highlight the relation between complexity, chaos and scrambling in chaotic systems. We conclude with a discussion of open problems and future directions. This article was written for the special issue of EPJ-C Frontiers in Holographic Duality.

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“…7.1], it was shown that the function M SU(2) ∋ g → diam(SU(2), g) σ 2 (g) is bounded on both sides by positive numbers. On the other hand, the articles [PS16] and [Po18] obtain explicit expressions for diam(SU(2), g) and diam(SO(3), g) provided that at least two elements in {σ 1 (g), σ 2 (g), σ 3 (g)} coincide. Each of these metrics is homothetic to a Berger 3-sphere.…”

Section: Introductionmentioning

confidence: 99%

Diameter and Laplace eigenvalue estimates for left-invariant metrics on compact Lie groups

Lauret

2020

Preprint

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Let G be a compact connected Lie group of dimension m. Once a bi-invariant metric on G is fixed, left-invariant metrics on G are in correspondence with m × m positive definite symmetric matrices. We estimate the diameter and the smallest positive eigenvalue of the Laplace-Beltrami operator associated to a left-invariant metric on G in terms of the eigenvalues of the corresponding positive definite symmetric matrix. As a consequence, we give partial answers to a conjecture by Eldredge, Gordina and Saloff-Coste; namely, we give large subsets S of the space of left-invariant metrics M on G such that there exists a positive real number C depending on G and S such that λ 1 (G, g) diam(G, g) 2 ≤ C for all g ∈ S. The existence of the constant C for S = M is the original conjecture.

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Cut locus of a left invariant Riemannian metric on SO3 in the axisymmetric case

Cited by 20 publications

References 7 publications

Conjugate points for systems of second-order ordinary differential equations

Conjugate points for systems of second-order ordinary differential equations

Quantum computational complexity from quantum information to black holes and back

Diameter and Laplace eigenvalue estimates for left-invariant metrics on compact Lie groups

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