of a differential quadratic form
An abbreviated notation for the expression
The symbols are called the Christoffel symbols of the first kind, in contrast to the Christoffel symbols of the second kind, , defined by
where is defined as follows:
These symbols were introduced by E.B. Christoffel in 1869.
Let , , be a linear connection on a manifold , where denotes the space of vector fields on . Let be a chart of . Then on , is completely determined by , where are coordinates on . The Christoffel symbols of the connection are now given by
It is important to note that the are not the components of a tensor field. In fact if the denote the Christoffel symbols of with respect to a second set of coordinates on , then
Let now be the Riemannian connection (cf. Riemannian geometry) defined by a (local) Riemannian metric . Then the Christoffel symbols of this quadratic differential form are those of the connection . I.e.,
so that indeed
where the are the Christoffel symbols of the second kind of the quadratic differential form as defined above.
|[a1]||S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963) pp. Chapt. 4|
|[a2]||R.S. Millman, G.D. Parker, "Elements of differential geometry" , Prentice-Hall (1977) pp. Chapt. 7|
Christoffel symbol. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Christoffel_symbol&oldid=18161