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:

    Whatever may be our just grievances in the southern states, it is fitting that we acknowledge that, considering their poverty and past relationship to the Negro race, they have done remarkably well for the cause of education among us. That the whole South should commit itself to the principle that the colored people have a right to be educated is an immense acquisition to the cause of popular education.
    Fannie Barrier Williams (1855–1944)

    Artists have a double relationship towards nature: they are her master and her slave at the same time. They are her slave in so far as they must work with means of this world so as to be understood; her master in so far as they subject these means to their higher goals and make them subservient to them.
    Johann Wolfgang Von Goethe (1749–1832)