1. LOGO Abstract Vector bundles are utilized in the study of physical fields. The connection of a fiber bundle corresponds to the gauge potential, and the physical field corresponds to the curvature of the fiber bundle. In this study we consider the geometrical interpretation of fields interaction using the language of fiber bundle. We illustrate our geometrical picture by : Dirac fields, Nucleon field and Einstein field. 1. Adler, Ronald, (1975). Introduction To General Relativity Second Edition. McGraw-HILL Book Company, New York. 2. Artzy, Rafael, (1965). Linear geometry, Addison – Wesley Publishing Company, Inc. New York. 3. Beiser, Arthur, (1995).Theory and Problems of Applied Physics, third edition, McGraw-Hill, Inc. New York. 4. Bleecker, David, (1981). Gauge Theory and Variational Principle, Addison - Wesley, London. 5. Boothby, William M., (1975 ). An Introduction to Differentiable Manifolds and Riemannian Geometry, Acadami Press, New York. 6. Fadeev, L. D., Slavnov, A. A., (1991). Gauge Fields : Introduction to Quantum Theory, Westwiew Press, USA. 7. Felice, F. De., Clarke, C.J.S., (1990). Relativity on curved manifolds, Cambridge University Press, New York. 8. F. Gesztesy, Univ. of Missouri, Foundations of Physics, Vol. 23, No. 7. 24 Mar 2011.The Dirac Equation,Springer- Verlag, New York. 9. Lee, H. C., (1984). An introduction to Kaluza ــKlein theories, world scientific,Canada. 10. ناجي. أسامة زيد ابراهيم ، ( 1998). مقدمة في ميكانيا الكم، الطبعة الأولى ، الدار الدولية للنشر والتوزيع ، القاهرة. 11. يس. محمد نبيل ، ويس. صلاح الدين البكري ، (2005). ميكانيكا الكم ، ط 1 ، دارالفكر العربي ، القاهرة. 12. يونس. عادل طه ،( 2010). النظرية النسبية الخاصة والعامة ، ط 1 ، دار الفكر العربي ، القاهرة. References We defined differentiable manifolds and their kinds, homogeneous physical spaces. We defined the Principal Fiber Bundle (PFB), the connection. We defined the – valued differential forms and the curvature of vector space. We made a one to one correspondence between the special linear group of transformations and Lorentz group to form a principal fiber bundle above which we studied the spin curvature (spinor) and deduced the global Dirac equation for the free electron. We set up the bundle splicing between Electroweak interaction and the electromagnetic field equation and of the nucleons with Yang-Mills field and deduced their global Dirac equation and global Lagrange equation respectively. We made a global tensor analysis and finding the Lie derivative of the Lagrangian, We proved its Lagrange equation and deduced the inhomogeneous field equation. Methods and Materials We found from Dirac assumptions for the spin curvature λ(p A) = λ(p)  (A) that this curvature is same as Maxwell inhomogeneous equation [ . j = d p / d t ] that changing charge in a closed cylindrical tube gives us the induced current.This leads to unification of weak interactions and electromagnetic ones. This encouraged us to splice them to find the global inhomogeneous Maxwell equation which is unified with Einstien field equation under specific situation (changing the bundle metric) in chapter five. Conclusions Introduction In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under a continuous group of local transformations. Many powerful theories in physics are described by Lagrangian which are invariant under some symmetry transformation groups. When they are invariant under a transformation identically performed at every point in the space in which the physical processes occur, they are said to have a local symmetry. The requirement of global symmetry, the cornerstone of gauge theories, is a stricter constraint. In fact, a global symmetry is just a local symmetry whose group's parameters are fixed in space-time. Gauge theories are important as the successful field theories explaining the dynamics of symmetry particles. Quantum electrodynamics is an abelian gauge theory with the symmetry group U(1 ) and has one gauge field, the electromagnetic four potential, with the photon being the gauge boson. The Standard Modal is a non- abelian gauge theory with the symmetry group U(1)× SU(2 ))× SU(3 ) and has a total of twelve gauge bosons: the photon, three weak bosons and eight gluons. 1.The researcher recommends to establish a mathematical model for the spin curvature to be suitable for gravitation field. 2.The researcher recommends those people who call for unification of the four forces in nature to generalize the spin curvature to be suitable for gravitation field equation. 3.The researcher recommends physicians to treat particles inside atoms as a vibration not distinct points, this leads to improve the idea of super string theory. Researches’ Center : Sudan University of Science and Technology College : Sciences Department : Mathematics Research Title : The Geometrical Formulation of Gauge Theories Researcher Name : Dr. Omima Elnour Saeed Recommendations LOGO