Biharmonic hypersurfaces in Riemannian manifolds

We study biharmonic hypersurfaces in a generic Riemannian manifold. We first derive an invariant equation for such hypersurfaces generalizing the biharmonic hypersurface equation in space forms studied in \cite{Ji2}, \cite{CH}, \cite{CMO1}, \cite{CMO2}. We then apply the equation to show that the generalized Chen's conjecture is true for totally umbilical biharmonic hypersurfaces in an Einstein space, and construct a (2-parameter) family of conformally flat metrics and a (4-parameter) family of multiply warped product metrics each of which turns the foliation of an upper-half space of \mathhbb{R}^m by parallel hyperplanes into a foliation with each leave a proper biharmonic hypersurface. We also characterize proper biharmonic vertical cylinders in $S^2\times \mathbb{R}$ and $H^2\times \mathbb{R}$.Comment: 16 pages with a correction to Theorem 3.

Multi-Photon Interference and Temporal Distinguishability of Photons

A number of recent interference experiments involving multiple photons are reviewed. These experiments include generalized photon bunching effects, generalized Hong-Ou-Mandel interference effects and multi-photon interferometry for demonstrations of multi-photon de Broglie wavelength. The multi-photon states used in these experiments are from two pairs of photons in parametric down-conversion. We find that the size of the interference effect in these experiments, characterized by the visibility of interference pattern, is governed by the degree of distinguishability among different pairs of photons. Based on this discovery, we generalize the concept of multi-photon temporal distinguishability and relate it to a number of multi-photon interference effects. Finally, we make an attempt to interpret the coherence theory by the multi-photon interference via the concept of temporal distinguishability of photons.Comment: fixed figures 4,5,