Contents -- Preface -- V. V. Balashchenko Invariant Structures Generated by Lie Group Automorphisms on Homogeneous Spaces -- 1. Introduction -- 2. Homogeneous regular -spaces -- 3. Canonical affinor structures on regular -spaces -- 4. The algebra A for regular -spaces -- 5. The classes of regular -spaces -- 6. Linear subspaces generated by -- 7. Canonical structures P,J,f,h and Riemannian metrics -- 8. Metric f-structures in generalized Hermitian geometry -- 9. Invariant NKf-structures on naturally reductive homogeneous spaces -- 10. Canonical NKf-structures on regular spaces -- 11. Examples -- 11.1. Riemannian 4- and 5-symmetric spaces -- 11.2. The spheres S2, S5, S6 -- 11.3. Homogeneous -spaces of order 8 -- 11.4. The 6-dimensional generalized Heisenberg group -- 11.5 The group of hyperbolic motions of the plane R2 -- 11.6. The flag manifold SU(3)/Tmax -- 11.7. The regular -space with non-semisimple -- References -- M. B. Banaru On Kenmotsu Hypersurfaces in a Six-Dimensional Hermitian Submanifold of Cayley Algebra -- 1. Introduction -- 2. Preliminaries -- 3. The Main Results -- References -- N. Blazic and S. Vukmirovic Para-hypercomplex Structures on a Four-Dimensional Lie Group -- 1. Introduction -- 2. Preliminaries -- 3. Lie algebras admitting a para-hypercomplex structure -- 3.1. Case when g has a non-trivial center -- 3.2. Case of solvable Lie algebra g and dimg' 2. -- Acknowledgments -- References -- A. V. Bolsinov and B. Jovanovic Integrable Geodesic Flows on Riemannian Manifolds: Construction and Obstructions -- 1. Introduction -- 1.1. Geodesics on Riemannian manifolds -- 1.2. Integrable geodesic flows -- 1.3. Statement of the problem -- 2. Classical examples of integrable geodesic flows -- 3. Topological obstructions to integrability -- 3.1. Case of two-dimensional surfaces.

3.2. Topological obstructions in the case of non simply connected manifolds -- 3.3. Topological entropy and integrability of geodesic flows -- 4. Counterexamples -- 5. Geodesic flows on homogeneous spaces and bi-quotients of Lie groups -- 5.1. Non-commutative integrability -- 5.2. Integrable geodesic flows on G / H and K\G/H -- 6. Complete commutative algebras on T*(G/H) -- 6.1. Mishchenko-Fomenko conjecture -- 6.2. Integrable pairs -- 7. Integrable deformations of normal metrics -- 8. Methods and examples -- 8.1. Argument shift method -- 8.2. Chains of subalgebras -- 8.3. Generalized chain method -- 9. Integrability and reduction -- 10. Geodesic flows on the spheres -- Acknowledgments -- References -- M. DjoriC and M. Okumura CR Submanifolds of Maximal CR Dimension in Complex Space Forms and Second Fundamental Form -- 1. Introduction -- 2. CR submanifolds of maximal CR dimension of complex space forms -- 3. CR submanifolds satisfying h(FX, Y ) + h(X, F Y ) = 0. -- 4. CR submanifolds satisfying h(FX, Y ) - h(X, F Y ) = 0 -- References -- M. Carmen Domingo-Juan and V. Miquel Pappus-Guldin's Formulae versus Weyl's Tube Formula: Old and Recent Results -- 1. Introduction -- 2. Motion along a curve -- 2.1. Definitions -- 2.2. Motions as curves in SO(n) and o(n ) -- 2.3. Moment respect to a geodesic hyperplane and center of mass -- 2.4. The volume of a domain obtained by the motion along a curve -- 2.5. The volume of a hypersurface obtained by the motion along a curve -- 3. Motion along a submanifold -- 3.1. Definitions -- 3.2. q-center of mass -- 3.3. The volume of D -- 3.4. Examples of q-symmetric domains -- 3.5. The volume of C -- 4. Holomorphic motions along complex curves -- 4.1. Definition of holomorphic and Frenet holomorphic motion -- 4.2. The volume of D for holomorphic motions -- 4.3. Bounds for vol(D) when n > 2 and P admits a Frenet motion.

4.4. Symmetries on Dp, which make (4.3) true for any motion -- References -- B. Dragovich Non-Archimedean Geometry and Physics on Adelic Spaces -- 1. Introduction -- 2. Non-Archimedean Geometry and p-Adic Numbers -- 3. Adeles and Their Functions -- 4. Quantum Mechanics on Adelic Spaces -- 5. Adelic Quantum Cosmology -- 6. Adelic String/M-theory -- 7. Concluding Remarks -- Acknowledgements -- References -- B. Dragovich and Z. Rakic Lagrangian Aspects of Quantum Dynamics on a Noncommutative Space -- 1. Introduction -- 2. Quadratic Lagrangians -- 2.1. Classical case -- 2.2. Noncommutative case -- 2.3. Linearization and some examples -- 3. Concluding Remarks -- Acknowledgements -- References -- M. Fernandez-Lopez, E. Garcia-Rio and D. N. Kupeli The Mobius Equation: A Local Analytical Characterization of Twisted Product Structures -- 1. Introduction -- 2. Preliminaries -- 2.1. Twisted and warped product structures -- 2.2. Second fundamental form of a map -- 3. Mobius equation associated to a map -- 4. Decomposition results -- 5. Mobius equation and affine maps -- References -- M. Fernandez, V. Munoz and J. A . Santisteban Symplectically Aspherical Manifolds with Nontrivial 2 and with No Kahler Metrics -- 1. Introduction -- 2. Preliminaries -- 3. The manifolds M2(n+1)(k) -- 4. Symplectically aspherical manifolds with nontrivial 2 and with no Kahler metrics -- References -- A. T. Fomenko and P. V. Morozov Some New Results in Topological Classification of Integrable Systems in Rigid Body Dynamics -- 1. Integrable Hamiltonian systems on symplectic manifolds -- 1.1. Completely integrable Hamiltonian systems -- 1.2. Liouville's theorem. -- 1.3. Equivalence relations of integrable Hamiltonian systems -- 2. Fomenko-Zieschang invariants of integrable Hamiltonian systems with two degrees of freedom -- 2.1. Isoenergy surfaces.

2.2. Critical sets on isoenergy surfaces -- 2.3. Neighborhoods of singular leaves on the isoenergy surfaces -- 2.4. Gluing matrices -- 3. Some new results of Liouville classification in rigid body dynamics -- 3.1. The phase space -- 3.2. Classical cases of integrability: Euler, Lagrange, Kovalevskaya -- 3.3. Liouville classification of the Clebsch case -- 3.4. Marked molecules of the Sokolov case -- References -- F. Gavarini The Crystal Duality Principle: From General Symmetries to Geometrical Symmetries -- 1. Introduction -- 2. Notation and terminology -- 2.1. Algebras, coalgebras, and the whole zoo. -- 2.2. Function algebras. -- 2.3. Enveloping algebras and symmetric algebras. -- 2.4. Filtrations. -- 3. Connecting functors on (co)augmented (co)algebras -- 4. Connecting and crystal functors on bialgebras and Hopf algebras -- 5. Deformations I - Rees algebras, Rees coalgebras, etc. -- 5.1. Filtrations and "Rees objects". -- 5.2. Connecting functors and Rees modules. -- 5.3. The bialgebra and Hopf algebra case. -- 6. Deformations II - from Rees bialgebras to quantum groups -- 7. Poisson duality and the Crystal Duality Principle -- Acknowledgements -- References -- Z. Hu and H. Li Willmore Submanifolds in a Riemannian Manifold -- 1. Introduction -- 2. Submanifolds in Riemannian manifold -- 3. Euler-Lagrange equation of the Willmore functional -- 4. Examples of Willmore submanifolds in Sn+P(1) -- 5. Willmore complex submanifolds in a complex space form -- 6. Willmore Lagrangian submanifolds in a complex space form -- Acknowledgments -- References -- Dj. Kadijevich Some Aspects of Visualizing Geometric Knowledge: Possibilities, Findings, Further Research -- 1. To visualize geometric knowledge or not? -- 2. Two tools for computer-based visualizations -- 2.1. Dynamic geometry environments -- 2.2. Java applets -- 3. Research outcomes and further directions.

4. Coda -- References -- V. P. Karassiov Dual Algebraic Pairs and Polynomial Lie Algebras in Quantum Physics: Foundations and Geometric Aspects. -- 1. Introduction -- 2. Dual algebraic pairs and polynomial Lie algebras in multiboson physics: a general analysis -- 3. Dual algebraic pairs in action: applications in polarization and nonlinear quantum optics -- 4. Conclusion -- Acknowledgments -- References -- E. Malkowsky and V. Velickovic Visualisation and Animation in Differential Geometry -- 1. Introduction and Notations -- 2. The General Concept and Main Principles of Our Software -- 2.1. Separation of geometry and drawing -- 2.2. Line graphics -- 2.3. Independent visibility check -- 3. The Procedure Check -- 4. Contour -- 5 . Curves -- 6. Surfaces -- 6.1. The normal and principal curvature -- 6.2. The geodesic curvature -- 7. Some Animations -- References -- G. V. Nosovskiy Computer Gluing of 2D Projective Images -- 1. Introduction -- 2. A method of recognition of conjugate points, based on the distribution of shifts between bounded patterns -- 3. The stability of determination of the projective mapping using a configuration of conjugate points -- References -- A . A . Oshemkov On the Topological Structure of the Set of Singularities of Integrable Hamiltonian Systems -- 1. Introduction -- 2. Nondegenerate singularities -- 3. Gauss-Bonnet formula for complex vector bundles -- 4. Some properties of the complex of singularities of integrable Hamiltonian systems -- References -- Translators' Notes: -- Th. Yu. Popelensky and Yu. P. Solovjev Lie-Cartan Pairs and Characteristic Classes in Noncommutative Geometry -- 1. Introduction -- 2. Characteristic classes of complex vector bundles -- 3. Characteristic classes of projective modules over an associative algebra -- 4. Mishchenko-Solovjev-Zhuraev construction -- 5. Z/2-graded Lie-Cartan pairs.

Appendix: The trace of an endomorphism of a projective module.