Symbolic tensor calculus on manifolds
                                                      Éric Gourgoulhon
                                            Laboratoire Univers et Théories (LUTH)
                                  CNRS / Observatoire de Paris / Université Paris Diderot
                                               Université Paris Sciences et Lettres
                                                      92190 Meudon, France
                                         http://sagemanifolds.obspm.fr/jncf2018/
                                    Journées Nationales de Calcul Formel
                            Centre International de Rencontres Mathématiques
                                                Luminy, Marseille, France
                                                    22-26 January 2018
              Éric Gourgoulhon                   Symbolic tensor calculus on manifolds                JNCF, 25 Jan 2018        1 / 38
     Outline
    1    Introduction
    2    Smooth manifolds
    3    Scalar fields
    4    Vector fields
    5    Tensor fields
    6    Conclusion and perspectives
                              http://sagemanifolds.obspm.fr/jncf2018/
              Éric Gourgoulhon                   Symbolic tensor calculus on manifolds                JNCF, 25 Jan 2018        2 / 38
                                                      Introduction
     Outline
    1    Introduction
    2    Smooth manifolds
    3    Scalar fields
    4    Vector fields
    5    Tensor fields
    6    Conclusion and perspectives
              Éric Gourgoulhon                   Symbolic tensor calculus on manifolds                JNCF, 25 Jan 2018        3 / 38
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      On pseudo-Riemannian manifolds:
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              musical isomorphisms
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Levi-Civita connection
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              curvature tensor
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              Hodge duality
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              ...
                                                      Introduction
     What is tensor calculus on manifolds?
    By tensor calculus it is usually meant
            arithmetics of tensor fields
            tensor product, contraction
            (anti)symmetrization
            Lie derivative along a vector field
            pullback and pushforward associated to a smooth manifold map
            exterior calculus on differential forms
            ...
              Éric Gourgoulhon                   Symbolic tensor calculus on manifolds                JNCF, 25 Jan 2018        4 / 38