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.

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.

scholarly journals DIRAC OPERATOR IN MATRIX GEOMETRY

2005 ◽  
Vol 02 (02) ◽  
pp. 227-264 ◽  
Author(s):  
IVAN G. AVRAMIDI
Keyword(s):  
Dirac Operator ◽  
Heat Kernel ◽  
Linear Combination ◽  
Differential Operators ◽  
Spectral Invariants ◽  
First Order ◽  
Partial Differential Operators ◽  
Laplace Type ◽  
Hilbert Action

We review the construction of the Dirac operator and its properties in Riemannian geometry, and show how the asymptotic expansion of the trace of the heat kernel determines the spectral invariants of the Dirac operator and its index. We also point out that the Einstein–Hilbert functional can be obtained as a linear combination of the first two spectral invariants of the Dirac operator. Next, we report on our previous attempts to generalize the notion of the Dirac operator to the case of Matrix Geometry, where, instead of a Riemannian metric there is a matrix valued self-adjoint symmetric two-tensor that plays a role of a "non-commutative" metric. We construct invariant first-order and second-order self-adjoint elliptic partial differential operators, which can be called "non-commutative" Dirac operators and non-commutative Laplace operators. We construct the corresponding heat kernel for the non-commutative Laplace type operator and compute its first two spectral invariants. A linear combination of these two spectral invariants gives a functional that can be considered as a non-commutative generalization of the Einstein–Hilbert action.

scholarly journals ON EIGENVALUE ESTIMATES FOR THE SUBMANIFOLD DIRAC OPERATOR

2002 ◽  
Vol 13 (05) ◽  
pp. 533-548 ◽  
Author(s):  
NICOLAS GINOUX ◽  
BERTRAND MOREL
Keyword(s):  
Dirac Operator ◽  
Lower Bounds ◽  
Dirac Operators ◽  
Extrinsic Curvature ◽  
Eigenvalue Estimates ◽  
Killing Spinors ◽  
Spinor Fields ◽  
Twisted Dirac Operators

We give lower bounds for the eigenvalues of the submanifold Dirac operator in terms of intrinsic and extrinsic curvature expressions. We also show that the limiting cases give rise to a class of spinor fields generalizing that of Killing spinors. We conclude by translating these results in terms of intrinsic twisted Dirac operators.

scholarly journals On Certain Proximities and Preorderings on the Transposition Hypergroups of Linear First-Order Partial Differential Operators

2014 ◽  
Vol 22 (1) ◽  
pp. 85-103 ◽  
Author(s):  
Jan Chvalina ◽  
Šárka Hošková-Mayerová
Keyword(s):  
Differential Operators ◽  
Partial Differential ◽  
First Order ◽  
Ordered Groups ◽  
Partial Differential Operators ◽  
Power Set

AbstractThe contribution aims to create hypergroups of linear first-order partial differential operators with proximities, one of which creates a tolerance semigroup on the power set of the mentioned differential operators. Constructions of investigated hypergroups are based on the so called “Ends-Lemma” applied on ordered groups of differnetial operators. Moreover, there is also obtained a regularly preordered transpositions hypergroup of considered partial differntial operators.

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