Riemannian Geometry and Geometric Analysis

Jost, Jürgen

doi:10.1007/978-3-662-04672-2

Cited by 701 publications

(1,017 citation statements)

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“…Proof One can apply a general Weitzenböck formula (see for example [12] or [11]) to prove the Proposition. For the convenience of the reader, we present a proof here.…”

Section: )mentioning

confidence: 99%

Dirac-harmonic maps

et al. 2006

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“…Proof One can apply a general Weitzenböck formula (see for example [12] or [11]) to prove the Proposition. For the convenience of the reader, we present a proof here.…”

Section: )mentioning

confidence: 99%

Dirac-harmonic maps

et al. 2006

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“…Its specific definition depends on the underlying space and on the algebra of functions on which it acts. For a generic smooth Riemannian manifold ðM; g μν Þ the LaplaceBeltrami operator acts on the algebra of smooth functions f ∈ C ∞ ðMÞ in the form [29]: Table I). The position of transition lines is only qualitative.…”

Section: The Laplace-beltrami Operatormentioning

confidence: 99%

Spectrum of the Laplace-Beltrami operator and the phase structure of causal dynamical triangulations

Clemente

D’Elia

2018

Phys. Rev. D

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We propose a new method to characterize the different phases observed in the nonperturbative numerical approach to quantum gravity known as causal dynamical triangulations. The method is based on the analysis of the eigenvalues and the eigenvectors of the Laplace-Beltrami operator computed on the triangulations: it generalizes previous works based on the analysis of diffusive processes and proves capable of providing more detailed information on the geometric properties of the triangulations. In particular, we apply the method to the analysis of spatial slices, showing that the different phases can be characterized by a new order parameter related to the presence or absence of a gap in the spectrum of the Laplace-Beltrami operator, and deriving an effective dimensionality of the slices at the different scales. We also propose quantities derived from the spectrum that could be used to monitor the running to the continuum limit around a suitable critical point in the phase diagram, if any is found.

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“…Note the (more than formal?) coincidence with the relation G = R − 1 2 tr[R]11 between the symmetric Einstein curvature tensor G and the symmetric Ricci curvature [13] tensor R in the Einstein field equations of general relativity theory [6,32]. Taking matrix-norms on both sides of Curl…”

Section: Combination Of the Resultsmentioning

confidence: 99%