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By Hamilton W.R.

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J. Math. Phys. 48, 123518, 17 p. (2007) 4. : Pseudo-Riemannian 3-manifolds with prescribed distinct constant Ricci eigenvalues. Differ. Geom. Appl. 26, 419–433 (2008) 5. : Curvature homogeneous Lorentzian three-manifolds. Ann. Glob. Anal. Geom. 36, 1–17 (2009) 6. : The equation of prescribed Ricci curvature. Bull. Am. Math. Soc. 3, 701–704 (1980) 7. : Existence of metrics with prescribed Ricci curvature: local theory. Invent. Math. 65, 179–207 (1981) 8. : The Cauchy problem for Lorentz metrics with prescribed Ricci curvature.

Math. 6, 45–61 (2011) 3. : Einstein spaces which are mapped conformally on each other. Math. Ann. 94, 119–145 (1925) Local Structure of Self-Dual Gradient Yamabe Solitons 35 4. : Four-dimensional neutral signature self-dual gradient Ricci soliton. Indiana Univ. Math. J. (to appear) 5. : Osserman and conformally Osserman manifolds with warped and twisted product structure. Results Math. 52, 211–221 (2008) 6. : The Geometry of Walker Manifolds. Synthesis Lectures on Mathematics and Statistics, vol.

12) 7 The (1 + 3) Decomposition of Einstein Field Equations Based on the (1 + 3) decomposition of both the Ricci tensor field and the stressenergy-momentum tensor field from the previous section, we express in a simple and elegant form the Einstein Field Equations (EFE). Let the EFE given by (cf. [4, p. 1) where G is the Newton constant, Λ is the cosmological constant, and T is the trace of T . 2) (c) R00 = Φ 2 {4π G(ρ + 3p) − Λ}. 2), we obtain 1 (a) Rik − bi bk − (bi|k + bk|i ) + Φ −2 {Θik|0 + (Θ − Ψ )Θik } 2 − {4π G(ρ − p) + Λ}hik − 8π Gπik = 0, j j j j j (b) Θi|j − Θ|i + Θci − Θi cj + Φ 2 {ωi|j + ωi cj + 2ωi bj } + 8π GΦqi = 0, 1 (c) Θ|0 + σ 2 + Θ 2 − Ψ Θ 3 j − Φ 2 {b |j + b2 + Φ 2 ω2 + 4π G(ρ + 3p) − Λ} = 0, where we put b2 = bj bj .

### Additional Applications of the Thepry of Algebraic Quaternions by Hamilton W.R.

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