Complete Formulas for the Volumes of Tubes about Curves in a Riemannian Manifold

By using the Taylor expansions of the solutions of Jacobi equations, we obtain the complete formulas for the volumes of tubes about curves in a Riemannian manifold. This unifies the known results and simplifies the computations involved in this direction. In the special case of surfaces, we also obtain the corresponding complete formulas which generalize the known results.