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The current work first explores projective collineations on pseudo-Riemannian manifolds. Projective collineations, curvature collineations, and Ricci curvature collineations are examined in relation to one another. On warped product manifolds, the projective collineations of the form ζ = ζ 1 + ζ 2 are investigated. We scrutinize various inheritance aspects in projective collineations from warped product manifolds to its factor manifolds. This provides, for example, a partially negative solution to Besse's problem regarding the existence of Einstein warped product manifolds. Finally, Pseudo-Ricci symmetric space-times admitting projective collineations are investigated. [ABSTRACT FROM AUTHOR]

The real projective plane has been formalized in Isabelle/HOL by Timothy Makarios [13] and in Coq by Nicolas Magaud, Julien Narboux and Pascal Schreck [12]. Some definitions on the real projective spaces were introduced early in the Mizar Mathematical Library by Wojciech Leonczuk [9], Krzysztof Prazmowski [10] and by Wojciech Skaba [18]. In this article, we check with the Mizar system [4], some properties on the determinants and the Grassmann-Plücker relation in rank 3 [2], [1], [7], [16], [17]. Then we show that the projective space induced (in the sense defined in [9]) by ℝ3 is a projective plane (in the sense defined in [10]). Finally, in the real projective plane, we define the homography induced by a 3-by-3 invertible matrix and we show that the images of 3 collinear points are themselves collinear. [ABSTRACT FROM AUTHOR]

In this paper we study Moufang-Klingenberg planes M(A) defined over a local alternative ring A of dual numbers. We show that some collineations of M(A) preserve cross-ratio and thus establish a relation between harmonicity and harmonic position. [ABSTRACT FROM AUTHOR]