The setting of this question is an n-dimensional manifold M equipped with metric tensor gj and covariant derivative Va. We work in a local coordinate system X. Let C be a curve written as X (A) and let its tangent vector be Ti (a) Consider a vector V evaluated along the curve C and define AV := V (X (X+SX)) V (X(A)) where SX < < 1. Write AV in terms of the curve's tangent vector, SA and the derivatives of V with respect to the local coordinates. (b) Introduce the object DX Va = TidVa. := Write this quantity in terms of AV and SX. (c) When discussing the geodesic deviation equation we introduced the notation D DX V := TVV . Write the equation for parallel transport using this notation and also write the geodesic (or autoparallel) equation in this notation. (d) Motivate the introduction of the connection I using the object AV jk (e) Let U be a vector with length = U$U$, (0) which is parallelly transported along the curve C. Find a condition on gij such that is preserved under parallel transport.