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Paneitz Operator
In the mathematics, mathematical field of differential geometry, the Paneitz operator is a fourth-order differential operator defined on a Riemannian manifold of dimension ''n''. It is named after Stephen Paneitz, who discovered it in 1983, and whose preprint was later published posthumously in . In fact, the same operator was found earlier in the context of conformal supergravity by E. Fradkin and A. Tseytlin in 1982 (Phys Lett B 110 (1982) 117 and Nucl Phys B 1982 (1982) 157 ). It is given by the formula :P = \Delta^2 - \delta \left\d + (n-4)Q where Δ is the Laplace–Beltrami operator, ''d'' is the exterior derivative, δ is its formal adjoint, ''V'' is the Schouten tensor, ''J'' is the trace of the Schouten tensor, and the dot denotes tensor contraction on either index. Here ''Q'' is the scalar invariant :(-4, V, ^2+nJ^2+2\Delta J)/4, where Δ is the positive Laplacian. In four dimensions this yields the Q-curvature. The operator is especially important in conformal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
