Christoffel Symbols
Let ∇ be the connection of the Riemannian metric. Choose local coordinates and let be the Christoffel symbols with respect to these coordinates. The torsion freeness condition 2 is then equivalent to the symmetry
The definition of the Levi-Civita connection derived above is equivalent to a definition of the Christoffel symbols in terms of the metric as
where as usual are the coefficients of the dual metric tensor, i.e. the entries of the inverse of the matrix .
Read more about this topic: Levi-Civita Connection
Famous quotes containing the word symbols:
“If the Americans, in addition to the eagle and the Stars and Stripes and the more unofficial symbols of bison, moose and Indian, should ever need another emblem, one which is friendly and pleasant, then I think they should choose the grapefruit. Or rather the half grapefruit, for this fruit only comes in halves, I believe. Practically speaking, it is always yellow, always just as fresh and well served. And it always comes at the same, still hopeful hour of the morning.”
—Johan Huizinga (1872–1945)