The authors develop a complete local theory for CR embedded submanifolds of CR manifolds in a way which parallels the Ricci calculus for Riemannian submanifold theory. They define a normal tractor bundle in the ambient standard tractor bundle along the submanifold and show that the orthogonal complement of this bundle is not canonically isomorphic to the standard tractor bundle of the submanifold. By determining the subtle relationship between submanifold and ambient CR density bundles the authors are able to invariantly relate these two tractor bundles, and hence to invariantly relate the normal Cartan connections of the submanifold and ambient manifold by a tractor analogue of the Gauss formula. This also leads to CR analogues of the Gauss, Codazzi, and Ricci equations. The tractor Gauss formula includes two basic invariants of a CR embedding which, along with the submanifold and ambient curvatures, capture the jet data of the structure of a CR embedding. These objects therefore form the basic building blocks for the construction of local invariants of the embedding. From this basis the authors develop a broad calculus for the construction of the invariants and invariant differential operators of CR embedded submanifolds. The CR invariant tractor calculus of CR embeddings is developed concretely in terms of the Tanaka-Webster calculus of an arbitrary (suitably adapted) ambient contact form. This enables straightforward and explicit calculation of the pseudohermitian invariants of the embedding which are also CR invariant. These are extremely difficult to find and compute by more naïve methods. The authors conclude by establishing a CR analogue of the classical Bonnet theorem in Riemannian submanifold theory.
VH Universe, journey to the end. The VLORs agents discovered a treacherous plot against cross-humans. An army of unidentified soldiers, led by Lieutenant Ogdo, have been capturing humans with special abilities around the globe. A daring rescue, gaining new allies, and leading to a surprise visit of a powerful Xeiar wizard arriving at Neirolo Cavern.
There are three dimensions, The Namorant, the Isolated, and Darkeil (Hell). What happened to cause strange creatures from the Namorant Dimension to cross over to the Isolated Dimension, where planet Earth lies? Join a very charismatic female heroine, two young Xeiar wizards, one ordinary human, two new cross-human breeds, and two Vexion creatures on the start of their adventure. Together, they will uncover hidden secrets, grow as a team, and deal with countless foes from Xanpo’s mysterious past.
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