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This lecture is devoted on some recent developments in the area of dynamic geometry on time scales. It is introduced the  covariant differentiation on time scales and they are deducted some  properties.We define  manifold,  open sets and  differentiable maps on  manifolds on time scales.  They are introduced tangent vector,  vector field,  Lie brackets and  Riemannian connection on time scales.  It is deducted the  Koszul formula and then they are given expressions of the  Christoffel coefficients. The results are provided with applications in the area of dynamic equations on time scales.