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CHRISTOFFEL SYMBOLS OF THE FIRST AND SECOND KIND



Christoffel symbols. Christoffel symbols are shorthand notations for various functions associated with quadratic differential forms. The differential form is usually the first fundamental quadratic form of a surface. Each Christoffel symbol is essentially a triplet of three indices, i, j and k, where each index can assume values from 1 to 2 for the case of two variables, or from 1 to n in the case of a quadratic form in n variables. Associated with different combinations of values of the three indices are different functions.


Christoffel symbols of the first kind. For the quadratic differential form in two variables


ole.gif


the Christoffel symbols of the first kind are defined as


ole1.gif


where the indices i, j and k can each assume the values of either 1 or 2.


Examples


             ole2.gif


             ole3.gif


 

Other notations. Other notations, instead of [i j, k], are used. They include Cijk and Γijk and


             ole4.gif





Meaning for the First Fundamental Quadratic Form. Let us examine the meaning of these Christoffel symbols for the first fundamental quadratic form


3)        Edu2 + 2Fdudv + Gdv2 .


Let S be a simple surface element defined by the one-to-one mapping


            x = x(u, v)

            y = y(u, v)

            z = z(u, v)


of a region R of the uv-plane into xyz-space. Let ole5.gif


             ole6.gif


be the position vector to point P on the surface. Let us use subscript notation to denote partial differentiation i.e.


             ole7.gif



The first fundamental quadratic form is


ole8.gif



where:


             ole9.gif


             ole10.gif


             ole11.gif



The functions that correspond to the different possible combinations of values of the indices i, j and k of the Christoffel symbol are [as can be confirmed by plugging the index values into 1)]:



             ole12.gif

 

             ole13.gif


             ole14.gif


The equivalence of the E, F and G expressions in terms of the ole15.gif is easily confirmed by taking the indicated partial derivatives of

 

             ole16.gif


The source of the Christoffel symbol notation becomes clear with a slight change in notation. If we use an integer subscript notation for the partial derivatives where


             ole17.gif


then


             ole18.gif


             ole19.gif


             ole20.gif



The Christoffel symbols are symmetric in i and j, which means that


            [i j, k] = [j i, k] .


Also, by definition, gij = gji.

 



Christoffel symbols of the second kind. For the quadratic differential form in two variables


             ole21.gif


the Christoffel symbols of the second kind are defined as


            Γi j k = Ak1[i j, 1] + Ak2[i j, 2]


where


1] the indices i, j and k can each assume the values of either 1 or 2,


2] Aki = Cki


            where Cki is the cofactor of gki in the determinant


             ole22.gif


 

3]        [i j, k] are the Christoffel symbols of the first kind.



For the first fundamental quadratic form


3)        Edu2 + 2Fdudv + Gdv2 .


the functions that correspond to the different possible combinations of values of the indices i, j and k are


             ole23.gif



             ole24.gif




             ole25.gif



Also,


             ole26.gif



The symbols in this case are also given by the following formulas:



             ole27.gif



             ole28.gif


             ole29.gif



The Christoffel symbols of the second kind correspond to the coefficients in the following equations of Gauss, from which they have their origin:


             ole30.gif


ole31.gif is the unit normal to the surface at point P and L,M, N are the second fundamental coefficients.




Quadratic form in n variables. For a quadratic form in n variables


               ole32.gif  


as might be encountered in differential geometry investigations in n-dimensional space, we remark that Christoffel symbols are also defined. Here the indices i, j and k can each assume values 1, 2, ... , n. It is assumed that gij = gji for all i and j. The general definition for n variables reduces to formula 2) above for the case of two variables.



References.

1. Graustein. Differential Geometry.

2. Lipschutz. Differential Geometry. Chapter 9.

3. James/James. Mathematics Dictionary.

4. Eisenhart. A Treatise on the Differential Geometry of Curves and Surfaces.


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