Design of Analog CMOS Integrated Circuits solutions by Behzad Razavi

By Behzad Razavi

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9 (1992) 1271. M. Senovilla, Class. Quantum Grav. J. Chinea, Class. Quantum Grav. 10, 2539 (1993); A. García, Class. Quantum Grav. 11 (1994) L45. J. Ernst, Phys. Rev. 167 (1968) 1175. [12] D. Kramer and G. Neugebauer, Commun. Math. Phys. 10 (1968) 132. [13] D. Kramer, Class. Quantum Grav. 1 (1984) L3; Astron. Nach. 307 (1986) 309. J. J. Pareja, Class. Quantum Grav. 16 (1999) 3823. This page intentionally left blank NEW DIRECTIONS FOR THE NEW MILLENNIUM Frederick J. Ernst * FJE Enterprises. Potsdam, New York 13676, USA Keywords: Exact solutions, potentials formaliosm, Geroch group.

Specializing this theorem for a maximally symmetric hypersurface immersed in a space the corresponding metric decomposes as follows: where Greek indices run and Latin indices run over We shall demonstrate that for a maximally symmetric hypersurface which, at the same time, is a totally geodesic hypersurface, the function entering in (47) has to be a constant (different from zero), which by scaling transformations of the can be equated to 1, 6. TOTALLY GEODESIC HYPERSURFACES An hypersurface is called a totally geodesic hypersurface of an enveloping if all the geodesics of are geodesics of For totally geodesic hypersurfaces there exists the following theorem, stated in the Eisenhart’s textbook [2], E183: A necessary and sufficient condition that a admits a family of totally geodesic hypersurfaces is that its fundamental form be reducible to where are independent of and is any function of and In comparison with the Eisenhart’s coordinates, we have replaced correspondingly by and by Notice that in (48), in front of stands the unit as a factor.

The maximal symmetry of a family of subspaces imposes very strong constraints on the metric of the whole space. The main theorem on maximally symmetric spaces, see W396, can be formulated as follows: It is always possible to choose the coordinates of the maximally symmetric subspaces and the coordinates of the subspaces so that the metric of the whole space is given by where and are functions of the alone, and are functions of the alone, thus is by itself the metric of an maximally symmetric space.

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