The riemannian geometry of holomorphic curves

AbstractIn this paper we study an analog of minimal surfaces called Weyl-minimal surfaces in conformal manifolds with a Weyl connection (M4, c, D). We show that there is an Eells-Salamon type correspondence between nonvertical 𝒥-holomorphic curves in the weightless twistor space and branched Weyl-minimal surfaces. When (M, c, J) is conformally almost-Hermitian, there is a canonical Weyl connection. We show that for the canonical Weyl connection, branched Weyl-minimal surfaces satisfy the adjunction inequality\chi \left( {{T_f}\sum } \right) + \chi \left( {{N_f}\sum } \right) \le \pm {c_1}\left( {f*{T^{\left( {1,0} \right)}}M} \right).The ±J-holomorphic curves are automatically Weyl-minimal and satisfy the corresponding equality. These results generalize results of Eells-Salamon and Webster for minimal surfaces in Kähler 4-manifolds as well as their extension to almost-Kähler 4-manifolds by Chen-Tian, Ville, and Ma.

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BRST and Superfield Formalism—A Review

Universe ◽

10.3390/universe7080280 ◽

2021 ◽

Vol 7 (8) ◽

pp. 280

Author(s):

Loriano Bonora ◽

Rudra Prakash Malik

Keyword(s):

Riemannian Geometry ◽

Gauge Theories ◽

Original Material ◽

Comprehensive Description ◽

Higher Spin ◽

Gauge Anomalies

This article, which is a review with substantial original material, is meant to offer a comprehensive description of the superfield representations of BRST and anti-BRST algebras and their applications to some field-theoretic topics. After a review of the superfield formalism for gauge theories, we present the same formalism for gerbes and diffeomorphism invariant theories. The application to diffeomorphisms leads, in particular, to a horizontal Riemannian geometry in the superspace. We then illustrate the application to the description of consistent gauge anomalies and Wess–Zumino terms for which the formalism seems to be particularly tailor-made. The next subject covered is the higher spin YM-like theories and their anomalies. Finally, we show that the BRST superfield formalism applies as well to the N=1 super-YM theories formulated in the supersymmetric superspace, for the two formalisms go along with each other very well.

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Uniqueness of Curvature Measures in Pseudo-Riemannian Geometry

Journal of Geometric Analysis ◽

10.1007/s12220-021-00702-4 ◽

2021 ◽

Author(s):

Andreas Bernig ◽

Dmitry Faifman ◽

Gil Solanes

Keyword(s):

Riemannian Manifolds ◽

Riemannian Geometry ◽

Riemannian Space ◽

Space Forms ◽

Type Formula ◽

Isometric Embeddings ◽

Curvature Measures ◽

Riemannian Space Forms

AbstractThe recently introduced Lipschitz–Killing curvature measures on pseudo-Riemannian manifolds satisfy a Weyl principle, i.e. are invariant under isometric embeddings. We show that they are uniquely characterized by this property. We apply this characterization to prove a Künneth-type formula for Lipschitz–Killing curvature measures, and to classify the invariant generalized valuations and curvature measures on all isotropic pseudo-Riemannian space forms.

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A relativistic basis of the quantum theory.—II

Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character ◽

10.1098/rspa.1934.0126 ◽

1934 ◽

Vol 145 (855) ◽

pp. 645-656 ◽

Cited By ~ 1

Keyword(s):

Quantum Mechanics ◽

Wave Equation ◽

Quantum Theory ◽

General Expression ◽

Matrix Equation ◽

Covariant Derivative ◽

Riemannian Geometry ◽

Order Equation ◽

Second Order ◽

Space Curvature

The paper is a continuation of the last paper communicated to these 'Proceedings.' In that paper, which we shall refer to as the first paper, a more general expression for space curvature was obtained than that which occurs in Riemannian geometry, by a modification of the Riemannian covariant derivative and by the use of a fifth co-ordinate. By means of a particular substitution (∆ μσ σ = 1/ψ ∂ψ/∂x μ ) it was shown that this curvature takes the form of the second order equation of quantum mechanics. It is not a matrix equation, however but one which has the character of the wave equation as it occurred in the earlier form of the quantum theory. But it contains additional terms, all of which can be readily accounted for in physics, expect on which suggested an identification with energy of the spin.

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WEYL GEOMETRY AS CHARACTERIZATION OF SPACE-TIME

International Journal of Modern Physics Conference Series ◽

10.1142/s2010194511001176 ◽

2011 ◽

Vol 03 ◽

pp. 87-97 ◽

Cited By ~ 5

Author(s):

F. P. POULIS ◽

J. M. SALIM

Keyword(s):

General Relativity ◽

Scalar Field ◽

Riemannian Geometry ◽

Axiomatic Approach ◽

Space Time ◽

Weyl Geometry ◽

Test Particles ◽

Variational Formalism ◽

Physical Fields

Motivated by an axiomatic approach to characterize space-time it is investigated a reformulation of Einstein's gravity where the pseudo-riemannian geometry is substituted by a Weyl one. It is presented the main properties of the Weyl geometry and it is shown that it gives extra contributions to the trajectories of test particles, serving as one more motivation to study general relativity in Weyl geometry. It is introduced its variational formalism and it is established the coupling with other physical fields in such a way that the theory acquires a gauge symmetry for the geometrical fields. It is shown that this symmetry is still present for the red-shift and it is concluded that for cosmological models it opens the possibility that observations can be fully described by the new geometrical scalar field. It is concluded then that this reformulation, although representing a theoretical advance, still needs a complete description of their objects.

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