This investigation looks at the field of geometric analysis and its applications in mathematical physics. Geometric analysis integrates techniques from differential geometry, functional analysis, and partial differential equations to study geometric structures and the physical events that accompany them. This study aims to provide a comprehensive grasp of the basic concepts, methods, and results in geometric analysis that are relevant to applications in mathematical physics. Starting with connections, curvature, geodesics, Riemannian and symplectic manifolds, and manifolds, the research explains the fundamental concepts of differential geometry. These concepts form the foundation for comprehending the geometric properties of physical systems and their mathematical descriptions. The study then explores the mathematical techniques used in geometric analysis, such as harmonic analysis, variational calculus, and variational techniques. The investigation is concentrated on several important geometric analysis applications in mathematical physics. One crucial area is the study of geometric flows like the Ricci flow and mean curvature flow. Understanding the behaviour and growth of geometric constructions depends on these fluxes. Another application is the study of harmonic maps and minimum surfaces, which has a direct impact on the study of membrane theory and the modelling of physical systems. Examining geometric phases and the geometric aspects of quantum field theory, the inquiry also looks at the connection between geometric analysis and quantum mechanics.
Key words: Riemannian manifolds, functional analysis, differential geometry, partial differential equations, mathematical physics.
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