The setting of this question is an n-dimensional manifold M equipped with metric tensor gj and...

 

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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 X¹. Let C be a curve written as X (A) and let its tangent vector be Ti (a) Consider a vector Vª evaluated along the curve C and define AVª := Vª (X¹ (X+SX)) — Vª (X¹(A)) where SX << 1. Write AV in terms of the curve's tangent vector, SA and the derivatives of Vª 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 Vª := T¹V¡Vª . 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 AVª jk (e) Let U¹ 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. 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 X¹. Let C be a curve written as X (A) and let its tangent vector be Ti (a) Consider a vector Vª evaluated along the curve C and define AVª := Vª (X¹ (X+SX)) — Vª (X¹(A)) where SX << 1. Write AV in terms of the curve's tangent vector, SA and the derivatives of Vª 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 Vª := T¹V¡Vª . 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 AVª jk (e) Let U¹ 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.