Abstract
Abstract Differentiation of tensor components in a curved space must be handled with extra care. By adding another term (related to Christoffel symbols) to the ordinary derivative operator, we can form a “covariant derivative”; such a differentiation operation does not spoil the tensor property. The relation between Christoffel symbols and first derivatives of metric functions is re-established. Using the concept of parallel transport, the geometric meaning of covariant differentiation is further clarified. The curvature tensor for an n-dimensional space is derived by the parallel transport of a vector around a closed path. Symmetry and contraction properties of the Riemann curvature tensor are considered. We find just the desired tensor needed for GR field equation.
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