Differentialgeometrie des isotropen Raumes, V. Zur Theorie der Eilinien

Strubecker, K.

doi:10.1007/bf01181547

Cited by 21 publications

(26 citation statements)

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“…Isotropic geometry has been systematically developed by Strubecker [36][37][38] in the 1940s. It is based on a simple semi-Riemannian metric and briefly surveyed in Section 2.1.…”

Section: And 33)mentioning

confidence: 99%

Laguerre minimal surfaces, isotropic geometry and linear elasticity

2008

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Laguerre minimal (L-minimal) surfaces are the minimizers of the energy (H 2 − K)/KdA. They are a Laguerre geometric counterpart of Willmore surfaces, the minimizers of (H 2 − K)dA, which are known to be an entity of Möbius sphere geometry. The present paper provides a new and simple approach to L-minimal surfaces by showing that they appear as graphs of biharmonic functions in the isotropic model of Laguerre geometry. Therefore, L-minimal surfaces are equivalent to Airy stress surfaces of linear elasticity. In particular, there is a close relation between L-minimal surfaces of the spherical type, isotropic minimal surfaces (graphs of harmonic functions), and Euclidean minimal surfaces. This relation exhibits connections to geometrical optics. In this paper we also address and illustrate the computation of L-minimal surfaces via thin plate splines and numerical solutions of biharmonic equations. Finally, metric duality in isotropic space is used to derive an isotropic counterpart to L-minimal surfaces and certain Lie transforms of L-minimal surfaces in Euclidean space. The latter surfaces possess an optical interpretation as anticaustics of graph surfaces of biharmonic functions.2000 Mathematics Subject Classification: 68U05, 53A40, 52C99, 51B15.Keywords differential geometry, Laguerre geometry, Laguerre minimal surface, isotropic geometry, linear elasticity, Airy stress function, biharmonic function, thin plate spline, geometrical optics.

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“…Isotropic geometry has been systematically developed by Strubecker [36][37][38] in the 1940s. It is based on a simple semi-Riemannian metric and briefly surveyed in Section 2.1.…”

Section: And 33)mentioning

confidence: 99%

Laguerre minimal surfaces, isotropic geometry and linear elasticity

2008

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“…In this case it is one of the Cayley-Klein geometries [5,46]. It is a well-researched geometry [35,36,37,38,39,40]. English textbooks are available [31,30,46].…”

Section: Pictorial Space As a Homogeneous Spacementioning

confidence: 99%

Gauge Fields in Pictorial Space

Koenderink¹,

Doorn²

2012

SIAM J. Imaging Sci.

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Abstract. "Pictorial space" is the mental structure that appears to be the scaffold for the visual awareness when looking "into" (as opposed to "at") a picture. Its structure differs from the "visual space" that is the scaffold for the visual awareness when looking into the scene in front of the observer. The structure of pictorial space has been probed empirically and explored theoretically. Here we propose a framework that allows one to handle cases that have been encountered empirically, but thus far have not been explored in a formal, geometrical setting. The framework allows one to handle many idiosyncrasies of human visual observers, as well as to characterize the (frequent) individual differences in a principled manner. This opens the door to a principled formalism of the structure (e.g., quality) of the pictorial spaces evoked by various methods of presentation, as required for applications.

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“…A degenerate metric prevents the standard constructions like covariant derivative and curvature. Manifolds endowed with degenerate metrics were studied by Moisil [41], Strubecker [60][61][62][63], Vrȃnceanu [64]. Notable is the work of Kupeli [37][38][39], which is limited to the constant signature case.…”

Section: Motivation and Related Advancesmentioning

confidence: 99%

On singular semi-Riemannian manifolds

Stoica

2014

Int. J. Geom. Methods Mod. Phys.

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Abstract. On a Riemannian or a semi-Riemannian manifold, the metric determines invariants like the Levi-Civita connection and the Riemann curvature. If the metric becomes degenerate (as in singular semi-Riemannian geometry), these constructions no longer work, because they are based on the inverse of the metric, and on related operations like the contraction between covariant indices.In this article we develop the geometry of singular semi-Riemannian manifolds. First, we introduce an invariant and canonical contraction between covariant indices, applicable even for degenerate metrics. This contraction applies to a special type of tensor fields, which are radical-annihilator in the contracted indices. Then, we use this contraction and the Koszul form to define the covariant derivative for radical-annihilator indices of covariant tensor fields, on a class of singular semi-Riemannian manifolds named radical-stationary. We use this covariant derivative to construct the Riemann curvature, and show that on a class of singular semi-Riemannian manifolds, named semi-regular, the Riemann curvature is smooth.We apply these results to construct a version of Einstein's tensor whose density of weight 2 remains smooth even in the presence of semi-regular singularities. We can thus write a densitized version of Einstein's equation, which is smooth, and which is equivalent to the standard Einstein equation if the metric is non-degenerate.

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Differentialgeometrie des isotropen Raumes, V. Zur Theorie der Eilinien

Cited by 21 publications

References 1 publication

Laguerre minimal surfaces, isotropic geometry and linear elasticity

Laguerre minimal surfaces, isotropic geometry and linear elasticity

Gauge Fields in Pictorial Space

On singular semi-Riemannian manifolds

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