Titlebook: A Perspective on Canonical Riemannian Metrics; Giovanni Catino,Paolo Mastrolia Book 2020 The Editor(s) (if applicable) and The Author(s),

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Information Transfer in Canonical Systems,In this first, introductory chapter we recall some important definitions and results of Riemannian Geometry, essentially following [1]. Although we assume the reader to be familiar with the general subject, as presented, e.g., in the standard references [109, 110, 15, 132, 78, 74], several computations and proofs will be provided in full detail.

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Economics of Trenchless Technology,In this chapter we introduce a second possible way to study canonical metrics on Riemannian manifolds, namely the one related to “Critical Metrics of Riemannian functionals” (CM, for short).

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https://doi.org/10.1007/978-1-4615-3058-9Bochner-Weitzenböck formulas for the Weyl tensor have been widely used in the last decades; to indicate just some of these works, focused on the study of Einstein manifolds and related structures, we mention those of Derdzinski [68], Singer [139], Hebey-Vaugon [93], Gursky [84, 86], Gursky-Lebrun [87], Yang [149] (see also the references therein).

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https://doi.org/10.1007/978-1-349-27433-8As in Chapter 4, we denote the spaces of Ricci solitons and of gradient Ricci solitons by ε. and ε. , respectively. A soliton X is . if X is a Killing vector field, or if △ f is parallel in the gradient case: clearly, a trivial Ricci soliton is an Einstein manifold.

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