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Lagrangian surfaces are a tool in the study of the geometry and topology of the ambient symplectic four-manifold.We will review the landmark result of M Gromov from 1985,where he proved that there are no embedded exact Lagrangian tori in \R^4.In analogy to this we then consider(necessarily exact)Lagrangian spheres in symplectic TS^2,which arise naturally in classical Differential Geometry.We will prove that the collection of such is locally a Banach manifold near sections with only one complex point.We will conclude by extracting corollaries in Differential Geometry and raise open problems in symplectic geometry of TS^2.