By Krishan L. Duggal, Bayram Sahin

This is often the 1st systematic account of the most ends up in the speculation of lightlike submanifolds of semi-Riemannian manifolds that have a geometrical constitution, equivalent to nearly Hermitian, nearly touch metric or quaternion K?hler. utilizing those buildings, the booklet offers attention-grabbing sessions of submanifolds whose geometry is particularly wealthy. The publication additionally comprises hypersurfaces of semi-Riemannian manifolds, their use commonly relativity and Osserman geometry, half-lightlike submanifolds of semi-Riemannian manifolds, lightlike submersions, display conformal submersions, and their purposes in harmonic maps. easy buildings and definitions are provided as initial history in each bankruptcy. The presentation explores functions and indicates a number of open questions. This self-contained monograph presents updated learn in lightlike geometry and is meant for graduate scholars and researchers simply getting into this box.

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21) it has been proved (see Kobayashi-Nomizu [263, page 13]) that the vector ﬁeld V generates a local ﬂow on M . If each integral curve of V is deﬁned on the entire real line, we say that V is a complete vector ﬁeld and it generates a global ﬂow on M . A set of local (resp. complete) integral curves is called a local congruence ( resp. congruence) of curves of V . Now we show how the ﬂow φ is used to transform any object, say Ω, on M into another one of the same type as Ω, with respect to a point transformation φt : xi → xi +tV i along an integral curve through xi .

Thus, λ + ν = f (t). Now integrating the ﬁrst equation and then using ∂r λ = −∂r ν, we get e2λ = e−2ν = 1− 2m r , where m is a positive constant. 4) takes the form ds2 = − 1 − 2m r dt2 + 1 − 2m r −1 dr2 + r2 dθ 2 + sin2 θdφ2 . 5) This solution is due to Schwarzschild for which M is the exterior Schwarzschild spacetime (r > 2m) with m and r as the mass and the radius of a spherical body. 3. 5) is singular at r = 0 and r = 2m. It is well known that r = 0 is an essential singularity and the singularity r = 2m can be removed by extending (M, g) to another manifold say (M , g ) as follows.