Christoffel Symbols - Relationship To Index-free Notation

Relationship To Index-free Notation

Let X and Y be vector fields with components and . Then the kth component of the covariant derivative of Y with respect to X is given by

Here, the Einstein notation is used, so repeated indices indicate summation over indices and contraction with the metric tensor serves to raise and lower indices:

Keep in mind that and that, the Kronecker delta. The convention is that the metric tensor is the one with the lower indices; the correct way to obtain from is to solve the linear equations .

The statement that the connection is torsion-free, namely that

is equivalent to the statement that —in a coordinate basis— the Christoffel symbol is symmetric in the lower two indices:

The index-less transformation properties of a tensor are given by pullbacks for covariant indices, and pushforwards for contravariant indices. The article on covariant derivatives provides additional discussion of the correspondence between index-free notation and indexed notation.

Read more about this topic:  Christoffel Symbols

Famous quotes containing the words relationship to and/or relationship:

    Women, because of their colonial relationship to men, have to fight for their own independence. This fight for our own independence will lead to the growth and development of the revolutionary movement in this country. Only the independent woman can be truly effective in the larger revolutionary struggle.
    Women’s Liberation Workshop, Students for a Democratic Society, Radical political/social activist organization. “Liberation of Women,” in New Left Notes (July 10, 1967)

    Christianity as an organized religion has not always had a harmonious relationship with the family. Unlike Judaism, it kept almost no rituals that took place in private homes. The esteem that monasticism and priestly celibacy enjoyed implied a denigration of marriage and parenthood.
    Beatrice Gottlieb, U.S. historian. The Family in the Western World from the Black Death to the Industrial Age, ch. 12, Oxford University Press (1993)