By Kouei Sekigawa, Vladimir S Gerdjikov, Stancho Dimiev

This booklet includes the contributions via the contributors within the 9 of a chain of workshops. in the course of the sequence of workshops, the participants are continuously aiming at larger achievements of experiences of the present subject matters in complicated research, differential geometry and mathematical physics and extra in any intermediate parts, with expectation of discovery of latest study instructions. in regards to the current one, it's important to say that, as well as the hot advancements of the conventional tendencies, many appealing and pioneering works have been offered and their effects have been contributed to the current quantity. The contents of this quantity consequently will offer not just major and necessary details for researchers in complicated research, differential geometry and mathematical physics (including their comparable areas), but in addition attention-grabbing arithmetic for non-specialists and a huge viewers. the current quantity comprises new advancements and tendencies within the reports on buildings of holomorphic Cliffordian features; the swelling buildings of minimum surfaces with greater genus in flat tori; the spectral homes of soliton equations on symmetric areas; new different types of shallow water waves defined through Camassa-Holm variety equations, the houses of pseudo-hermitian boson and fermion coherent states; fractals and chaos on orthorhombic lattices, or even an bold thought of a graph version for Kaehler manifolds with Kaehler magnetic fields.

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Extra info for Trends in Differential Geometry, Complex Analysis and Mathematical Physics: Proceedings of 9th International Workshop on Complex Structures, Integrability and Vector Fields

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It is easy to check that (∇∗ )∗ = ∇. In this paper, we consider generalizations of these conjugate connections. Let (M, g) be a semi-Riemannian manifold, ∇ an affine connection on M , and C a (0, 3)-tensor field on M . We define another affine connection ∇∗ by Xg(Y, Z) = g(∇X Y, Z) + g(Y, ∇∗ X Z) + C(X, Y, Z). (2) If the tensor C vanishes identically, then ∇∗ is the standard conjugate connection of ∇. Since the metric tensor g is symmetric, using twice Equation (2), we obtain the following proposition.

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In this paper, we consider generalizations of these conjugate connections. Let (M, g) be a semi-Riemannian manifold, ∇ an affine connection on M , and C a (0, 3)-tensor field on M . We define another affine connection ∇∗ by Xg(Y, Z) = g(∇X Y, Z) + g(Y, ∇∗ X Z) + C(X, Y, Z). (2) If the tensor C vanishes identically, then ∇∗ is the standard conjugate connection of ∇. Since the metric tensor g is symmetric, using twice Equation (2), we obtain the following proposition. 1. g((∇∗ )X Y − ∇X Y, Z) = C(X, Y, Z) − C(X, Z, Y ).

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