IMPROVED FORMS OF SOME VANISHING THEOREMS IN RIEMANNIAN SPIN GEOMETRY

2000 ◽  
Vol 11 (03) ◽  
pp. 291-304 ◽  
Author(s):  
THOMAS BRANSON ◽  
OUSSAMA HIJAZI
Keyword(s):  
Differential Operators ◽  
Infinite Sequence ◽  
Spin Manifold ◽  
Vanishing Theorems ◽  
Spin Geometry ◽  
First Order ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Higher Spin ◽  
Weyl Conformal Curvature Tensor

We improve the hypotheses on some vanishing theorems for first order differential operators on bundles over a Riemannian spin manifold. The improved hypotheses are uniform, in the sense that they are the same for each of an infinite sequence of bundles in each even dimension. They are also elementary, in the sense that they involve only the bottom eigenvalue of the Yamabe operator on scalars, and the pointwise action of the Weyl conformal curvature tensor on two-forms. In particular, they do not make reference to the higher spin bundles on which the conclusion holds.

Vanishing Theorems and Eigenvalue Estimates in Riemannian Spin Geometry

1997 ◽  
Vol 08 (07) ◽  
pp. 921-934 ◽  
Author(s):  
Thomas Branson ◽  
Oussama Hijazi
Keyword(s):  
Representation Theory ◽  
Dirac Operator ◽  
Differential Operators ◽  
Structure Group ◽  
Vanishing Theorems ◽  
Eigenvalue Estimates ◽  
Spin Geometry ◽  
First Order ◽  
Group Spin

We use the representation theory of the structure group Spin (n), together with the theory of conformally covariant differential operators, to generalize results estimating eigenvalues of the Dirac operator to other tensor-spinor bundles, and to get vanishing theorems for the kernels of first-order differential operators.

Some solutions of the Lanczos vacuum wave equation

1994 ◽  
Vol 447 (1931) ◽  
pp. 577-585 ◽  
Keyword(s):  
Wave Equation ◽  
Degrees Of Freedom ◽  
Vector Fields ◽  
Spin Coefficient ◽  
Independent Components ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Non Local ◽  
Weyl Conformal Curvature Tensor ◽  
Gauge Conditions

In general relativity the non-local part of the gravitational field is described by the 10 degrees of freedom of the Weyl conformal curvature tensor C abcd . In every space-time the Weyl field C abcd is derivable from a potential L abc which has at most 16 algebraically independent components reducing to 10 degrees of freedom when the six gauge conditions L ab s ; s = 0 are imposed. The potential L abc discov­ered by Lanczos was shown by Illge to have an extremely simple vacuum wave equation, namely, □ L abc ≡ g sm L abc ; s ; m = 0. Using tensor, spinor and spin-coefficient methods we give some solutions of this new vacuum wave equation in some spacetimes containing one or more preferred vector fields.

scholarly journals COMMENTS ON HIGHER-SPIN SYMMETRIES

2009 ◽  
Vol 06 (02) ◽  
pp. 285-342 ◽  
Author(s):  
XAVIER BEKAERT
Keyword(s):  
Lie Algebra ◽  
Differential Operators ◽  
Symmetric Tensor ◽  
Weak Field ◽  
Gauge Fields ◽  
Gauge Parameter ◽  
Gauge Transformations ◽  
First Order ◽  
Gauge Symmetries ◽  
Higher Spin

The unconstrained frame-like formulation of an infinite tower of completely symmetric tensor gauge fields is reviewed and examined in the limit where the cosmological constant goes to zero. By partially fixing the gauge and solving the torsion constraints, the form of the gauge transformations in the unconstrained metric-like formulation are obtained till first order in a weak field expansion. The algebra of the corresponding gauge symmetries is shown to be equivalent, at this order and modulo (unphysical) gauge parameter redefinitions, to the Lie algebra of Hermitian differential operators on ℝn, the restriction of which to the spin-two sector is the Lie algebra of infinitesimal diffeomorphisms.

scholarly journals Pseudosymmetry properties of generalised Wintgen ideal Legendrian submanifolds

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1209-1215
Author(s):  
Aleksandar Sebekovic ◽  
Miroslava Petrovic-Torgasev ◽  
Anica Pantic
Keyword(s):  
Curvature Tensor ◽  
Ricci Tensor ◽  
Space Forms ◽  
Equality Case ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Sasakian Space Forms ◽  
Legendrian Submanifold ◽  
Legendrian Submanifolds ◽  
Weyl Conformal Curvature Tensor

For Legendrian submanifolds Mn in Sasakian space forms ?M2n+1(c), I. Mihai obtained an inequality relating the normalised scalar curvature (intrinsic invariant) and the squared mean curvature and the normalised scalar normal curvature of M in the ambient space ?M (extrinsic invariants) which is called the generalised Wintgen inequality, characterising also the corresponding equality case. And a Legendrian submanifold Mn in Sasakian space forms ?M2n+1(c) is said to be generalised Wintgen ideal Legendrian submanifold of ?M2n+1(c) when it realises at everyone of its points the equality in such inequality. Characterisations based on some basic intrinsic symmetries involving the Riemann-Cristoffel curvature tensor, the Ricci tensor and the Weyl conformal curvature tensor belonging to the class of pseudosymmetries in the sense of Deszcz of such generalised Wintgen ideal Legendrian submanifolds are given.

scholarly journals Some invariants of conformal mappings of a generalized Riemannian space

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1465-1474
Author(s):  
Nenad Vesic
Keyword(s):  
Curvature Tensor ◽  
Riemannian Space ◽  
Affine Connection ◽  
Conformal Mappings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Necessary And Sufficient ◽  
Weyl Conformal Curvature Tensor

Invariants of conformal mappings between non-symmetric affine connection spaces are obtained in this paper. Correlations between these invariants and the Weyl conformal curvature tensor are established. Before these invariants, it is obtained a necessary and sufficient condition for a mapping to be conformal. Some appurtenant invariants of conformal mappings are obtained.

scholarly journals CONFORMALLY OSSERMAN MANIFOLDS AND CONFORMALLY COMPLEX SPACE FORMS

2004 ◽  
Vol 01 (01n02) ◽  
pp. 97-106 ◽  
Author(s):  
N. BLAŽIĆ ◽  
P. GILKEY
Keyword(s):  
Curvature Tensor ◽  
Complex Space ◽  
Jacobi Operator ◽  
Space Forms ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Weyl Curvature Tensor ◽  
Complex Space Forms ◽  
Weyl Conformal Curvature Tensor ◽  
Complex Projective

We characterize manifolds which are locally conformally equivalent to either complex projective space or to its negative curvature dual in terms of their Weyl curvature tensor. As a byproduct of this investigation, we classify the conformally complex space forms if the dimension is at least 8. We also study when the Jacobi operator associated to the Weyl conformal curvature tensor of a Riemannian manifold has constant eigenvalues on the bundle of unit tangent vectors and classify such manifolds which are not conformally flat in dimensions congruent to 2 mod 4.

scholarly journals On the intrinsic Deszcz symmetries and the extrinsic Chen character of Wintgen ideal submanifolds

2010 ◽  
Vol 41 (2) ◽  
pp. 109-116 ◽  
Author(s):  
S. Decu ◽  
M. Petrovic-Torgasev ◽  
A. Sebekovic ◽  
L. Verstraelen
Keyword(s):  
Ricci Curvature ◽  
Curvature Tensor ◽  
Real Space ◽  
Space Forms ◽  
Conformal Curvature ◽  
Conformal Curvature Tensor ◽  
Real Space Forms ◽  
Wintgen Ideal Submanifolds ◽  
Weyl Conformal Curvature Tensor

In this paper it is shown that all Wintgen ideal submanifolds in ambient real space forms are Chen submanifolds. It is also shown that the Wintgen ideal submanifolds of dimension $ >3 $ in real space forms do intrinsically enjoy some curvature symmetries in the sense of Deszcz of their Riemann--Christoffel curvature tensor, of their Ricci curvature tensor and of their Weyl conformal curvature tensor.

scholarly journals Higher-Spin Symmetries and Deformed Schrödinger Algebra in Conformal Mechanics

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Francesco Toppan ◽  
Mauricio Valenzuela
Keyword(s):  
Differential Operators ◽  
Similarity Transformation ◽  
Heisenberg Algebra ◽  
Dynamical Symmetries ◽  
First Order ◽  
Enveloping Algebra ◽  
Higher Spin ◽  
Invariant Differential ◽  
Order Invariant ◽  
Schrödinger Algebra

The dynamical symmetries of 1+1-dimensional Matrix Partial Differential Equations with a Calogero potential (with/without the presence of an extra oscillatorial de Alfaro-Fubini-Furlan, DFF, term) are investigated. The first-order invariant differential operators induce several invariant algebras and superalgebras. Besides the sl(2)⊕u(1) invariance of the Calogero Conformal Mechanics, an osp2∣2 invariant superalgebra, realized by first-order and second-order differential operators, is obtained. The invariant algebras with an infinite tower of generators are given by the universal enveloping algebra of the deformed Heisenberg algebra, which is shown to be equivalent to a deformed version of the Schrödinger algebra. This vector space also gives rise to a higher-spin (gravity) superalgebra. We furthermore prove that the pure and DFF Matrix Calogero PDEs possess isomorphic dynamical symmetries, being related by a similarity transformation and a redefinition of the time variable.

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