The Early Geometrical Works of Sophus Lie and Felix Klein

Rowe, David E.

doi:10.1016/b978-0-12-599661-7.50014-8

Cited by 18 publications

(16 citation statements)

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“…However it was Plücker, who first considered straight lines in R 3 as points of some space; he also found a convenient way to parametrize that space. In the second half of the XIX-th century this subject was very popular and studied, after Plücker, by many distinguished geometers; to mention Klein, Lie, Bianchi and Darboux only [42,2,14,23,31].…”

Section: Rectilinear Congruences and Quadrilateral Latticesmentioning

confidence: 99%

Transformations of quadrilateral lattices

Doliwa

Santini

Mañas

2000

Journal of Mathematical Physics

115

209

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Integrable lattices and their sublattices. II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices J. Math. Phys. 48, 113506 (2007) Motivated by the classical studies on transformations of conjugate nets, we develop the general geometric theory of transformations of their discrete analogs: the multidimensional quadrilateral lattices, i.e., lattices x:Z N →R M , NрM , whose elementary quadrilaterals are planar. Our investigation is based on the discrete analog of the theory of the rectilinear congruences, which we also present in detail. We study, in particular, the discrete analogs of the Laplace, Combescure, Lévy, radial, and fundamental transformations and their interrelations. The composition of these transformations and their permutability is also investigated from a geometric point of view. The deep connections between ''transformations'' and ''discretizations'' is also investigated for quadrilateral lattices. We finally interpret these results within the ‫ץ‬ formalism.

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Section: Rectilinear Congruences and Quadrilateral Latticesmentioning

confidence: 99%

Transformations of quadrilateral lattices

Doliwa

Santini

Mañas

2000

Journal of Mathematical Physics

115

209

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“…92 I'm indebted to Rowe (1994), 910, for drawing my attention to this and to Rowe (1989), Hawkins (1991), Hawkins (1989), and Hawkins (2000) chaps. 1 and 2 for guiding me through many intricate details.…”

Section: Abbe As Philosopher Riemannian and Student Of Foundationsmentioning

confidence: 98%

A Primer on Ernst Abbe for Frege Readers

Tappenden¹

2008

Can. j. philos. Suppl. vol.

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Setting out to understand Frege, the scholar confronts a roadblock at the outset: We just have little to go on. Much of the unpublished work and correspondence is lost, probably forever. Even the most basic task of imagining Frege's intellectual life is a challenge. The people he studied with and those he spent daily time with are little known to historians of philosophy and logic. To be sure, this makes it hard to answer broad questions like: ‘Who influenced Frege?’ But the information vacuum also creates local problems of textual interpretation. Say we encounter a sentence that may be read as alluding to a scientific dispute. Should it be read that way? To answer, we'd need to address prior questions. Is it reasonable to think Frege would be familiar with the issue? Deep or superficial familiarity? Would he expect his readers to catch the allusion? Can he be expected to anticipate certain objections? Can people he knows be expected to press those objections? A battery of such questions arise, demanding a richer understanding of Frege's environment.

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“…(36) Lie-dragging with the time translation and with ξ (2) forming the abelian subgroup leads to 1 γ ab = g ab,0 , 2 γ ab = g ab, 3 , and to the weakly Lie invariant metric (i.e., with L X 1 L X 1 g ab = 0, L X 2 L X 2 g ab = 0)…”

Section: Weak Spherical Symmetrymentioning

confidence: 99%

Weak Lie symmetry and extended Lie algebra

Goenner

2013

Journal of Mathematical Physics

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1The concept of weak Lie motion (weak Lie symmetry) is introduced through L ξ L ξ g ab = 0, (L ξ L ξ f = 0). Applications are given which exhibit a reduction of the usual symmetry, e.g., in the case of the the rotation group. In this context, a particular generalization of Lie algebras is found ("extended Lie algebras") which turns out to be an involutive distribution or a simple example for a tangent Lie algebroid. Riemannian and Lorentz metrics can be introduced on such an algebroid through an extended Cartan-Killing form. Transformation groups from non-relativistic mechanics and quantum mechanics lead to such tangent Lie algebroids and to Lorentz geometries constructed on them (1-dimensional gravitational fields).1 A summary of this article has been presented at the "90th Encounter between Mathematicians and Theoretical Physicists" at

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The Early Geometrical Works of Sophus Lie and Felix Klein

Cited by 18 publications

References 0 publications

Transformations of quadrilateral lattices

Transformations of quadrilateral lattices

A Primer on Ernst Abbe for Frege Readers

Weak Lie symmetry and extended Lie algebra

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