11. Analytic mappings induce pushforward maps on tangent spaces and pullback maps on cotangent spaces. 12. These points generate a tangent space of definite dimension " at " each point. 13. Locally ringed spaces have just enough structure to allow the meaningful definition of tangent spaces . 14. Tangent spaces to an n-dimensional smooth manifold are n-dimensional linear spaces.15. Further study of tangent measures and tangent spaces leads to the notion of a varifold. 16. The vectors in this tangent space are different from the vectors of the vector manifold. 17. The notion of approximate tangent spaces is very closely related to that of rectifiable sets. 18. Loosely speaking, rectifiable sets are precisely those for which approximate tangent spaces exist almost everywhere. 19. The tangent space at each event is a vector space of the same dimension as spacetime,. 20. For each, L ( v, q ) is a convex function of the tangent space .
