Generalized fundamental equations on the submanifolds of a manifoldESX n

A connection which is both Einstein and semisymmetric is called anES connection, and a generalizedn-dimensional Riemannian manifold on which the differential geometric structure is imposed by a unified field tensorg through anES connection is called ann-dimensionalES manifold and denoted byESX _n . We investigate some necessary and sufficient conditions for submanifolds ofESX _n to be also Einstein and derive the generalized fundamental equations on various submanifolds ofESX _n , such as generalized Gauss formulas, generalized Weingarten equations, and generalized Gauss-Codazzi equations. We employ the useful and powerful concept ofC-nonholonomic frame of reference, introduced in earlier work.     

Swallowtail porphyrins: synthesis, characterization and incorporation into porphyrin dyads

The incorporation of symmetrically branched tridecyl ("swallowtail") substituents at the meso positions of porphyrins results in highly soluble building blocks. Synthetic routes have been investigated to obtain porphyrin building blocks bearing 1-4 swallowtail groups. Porphyrin dyads have been synthesized in which the zinc or free base (Fb) porphyrins are joined by a 4,4'-diphenylethyne linker and bear swallowtail (or n-pentyl) groups at the nonlinking meso positions. The swallowtail-substituted Zn(2)- and ZnFb-dyads are readily soluble in common organic solvents. Static absorption and fluorescence spectra and electrochemical data show that the presence of the swallowtail groups slightly raises the energy level of the filled a(2u)(pi) HOMO. EPR studies of the pi-cation radicals of the swallowtail porphyrins indicate that the torsional angle between the proton on the alkyl carbon and p-orbital on the meso carbon of the porphyrin is different from that of a porphyrin bearing linear pentyl groups. Regardless, the swallowtail substituents do not significantly affect the photophysical properties of the porphyrins or the electronic interactions between the porphyrins in the dyads. In particular, time-resolved spectroscopic studies indicate that facile excited-state energy transfer occurs in the ZnFb dyad, and EPR studies of the monocation radical of the Zn(2)-dyad show that interporphyrin ground-state hole transfer is rapid.