Yi-Shi Duan — Page 4

arXiv1999

Generally Covariant Conservative Energy-Momentum for Gravitational Anyons

We obtain a generally covariant conservation law of energy-momentum for gravitational anyons by the general displacement transform. The energy-momentum currents have also superpotentials and are therefore identically conserved. It is shown that for Deser's solution and Clement's solution, the energy vanishes. The reasonableness of the definition of energy-momentum may be confirmed by the solution for pure Einstein gravity which is a limit of vanishing Chern-Simons coulping of gravitational anyons.

hep-th↗

arXiv1999

The relations among two transversal submanifolds and global manifold

In Riemann geometry, the relations among two transversal submanifolds and global manifold are discussed. By replacing the normal vector of a submanifold with the tangent vector of another submanifold, the metric tensors, Christoffel symbols and curvature tensors of the three manifolds are linked together. When the inner product of the two tangent vectors vanishes, some corollaries of these relations give the most important second fundamental form and Gauss-Codazzi equation in the conventional submanifold theory. As a special case, the global manifold is Euclidean is considered. It is pointed out that, in order to obtain the nonzero energy-momentum tensor of matter field in a submanifold, there must be the contributions of the above inner product and the other submanifold. In general speaking, a submanifold is closely related to the matter fields of the other submanifold through the above inner product. This conclusion is in agreement with the Kaluza-Klein theory and it can be applied to generalize the models of direct product of manifolds in string and D-brane theories to the more general cases --- nondirect product manifolds.

gr-qc↗