By Miguel Brozos-Vazquez, Peter B Gilkey, Stana Z Nikcevic
A vital quarter of research in Differential Geometry is the exam of the connection among the simply algebraic houses of the Riemann curvature tensor and the underlying geometric homes of the manifold. during this e-book, the findings of diverse investigations during this box of research are reviewed and provided in a transparent, coherent shape, together with the newest advancements and proofs. although many authors have labored during this sector in recent times, many basic questions nonetheless stay unanswered. Many reports commence through first operating in simple terms algebraically after which later progressing onto the geometric environment and it's been chanced on that many questions in differential geometry might be phrased as difficulties regarding the geometric consciousness of curvature. Curvature decompositions are vital to all investigations during this zone. The authors current a variety of effects together with the Singer-Thorpe decomposition, the Bokan decomposition, the Nikcevic decomposition, the Tricerri Vanhecke decomposition, the Gray-Hervella decomposition and the De Smedt decomposition. They then continue to attract applicable geometric conclusions from those decompositions.
The publication organizes, in a single coherent quantity, the result of learn accomplished via many alternative investigators during the last 30 years. whole proofs are given of effects which are frequently basically defined within the unique courses. while the unique effects are typically within the confident certain (Riemannian setting), the following the authors expand the implications to the pseudo-Riemannian atmosphere after which extra, in a fancy framework, to para-Hermitian geometry besides. as well as that, new effects are acquired to boot, making this a terrific textual content for a person wishing to additional their wisdom of the technological know-how of curvature.
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Extra resources for Geometric Realizations of Curvature
An almost hyper-complex structure J on a Riemannian manifold (M, g) is said to be almost hyper-Hermitian if each Ji is almost Hermitian; J is said to be hyper-Hermitian if J is an integrable hyper-complex structure. We say that (V, ⟨·, ·⟩, J , A) is a hyper-Hermitian curvature model if A ∈ R(V ) and if J is a hyper-Hermitian complex structure on (V, ⟨·, ·⟩). a) to this setting by deﬁning: τJ := τJ1 + τJ2 + τJ3 . 3 can be generalized to this setting to this setting [BrozosV´azquez et al. 2 Any hyper-Hermitian curvature model is geometrically realizable by an almost hyper-Hermitian manifold with τ and τJ constant.
1 (1) Let ϕ ∈ S 2 be a symmetric bilinear form. Set Aϕ (x, y, z, w) := ϕ(x, w)ϕ(y, z) − ϕ(x, z)ϕ(y, w). These tensors arise in the study of hypersurface theory; if ϕ is the second fundamental form of a hypersurface in ﬂat space, then the curvature tensor of the hypersurface is given by Aϕ . (2) Let ψ ∈ Λ2 be an anti-symmetric bilinear form. Set Aψ (x, y, z, w) := ψ(x, w)ψ(y, z) − ψ(x, z)ψ(y, w) − 2ψ(x, y)ψ(z, w). The study of the tensors Aψ arose in the original instance from the Osserman conjecture and related matters which are contained in [Garc´ıaR´ıo, Kupeli, and V´azquez-Lorenzo (2002)] and [Gilkey (2001)].
Because ξ1 = ker(π), the projection π : ξ1⊥ → ξ2 is 1-1 and onto. Consequently, π provides a natural module isomorphism between the modules ξ1⊥ and ξ2 . Assertion (3) now follows. 5, we may decompose ξ = η1 ⊕ · · · ⊕ ηℓ as the orthogonal direct sum of irreducible representations. Of course, the components ηi can be isomorphic. We group the isomorphic factors to express ξ = n1 ξ1 + · · · + nk ξk , where n1 factors of the ηi are isomorphic to ξ1 , where n2 factors of the ηi are isomorphic to ξ2 and so forth and where ξi is not isomorphic to ξj for i ̸= j.