Ngô Quốc Anh

January 23, 2010

RG: High order covariant derivatives

Filed under: Nghiên Cứu Khoa Học, Riemannian geometry — Ngô Quốc Anh @ 21:47

Today we talk about high order covariant derivative. Recall from this topic that covariant derivative \nabla acts on a (r,s)-tensor, the result is an (r+1,s)-tensor. For convenience, we mean

\displaystyle\alpha= \alpha _{{j_1} \cdots {j_r}}^{{k_1} \cdots {k_s}}\frac{\partial }{{\partial {x^{{k_1}}}}} \otimes\cdots\otimes \frac{\partial }{{\partial {x^{{k_s}}}}} \otimes d{x^{{j_1}}} \otimes\cdots\otimes d{x^{{j_r}}}.

Thus

\displaystyle\nabla :{C^\infty }({ \bigotimes\nolimits^{r,s}}M) \to {C^\infty }({ \bigotimes\nolimits^{r + 1,s}}M).

Precisely, for an (r,s)-tensor \alpha, one defines \nabla \alpha as the following

\displaystyle\left( {\nabla \alpha } \right)\left( {X,{Z_1},{Z_2},...,{Z_r}} \right) = \left( {{\nabla _X}\alpha } \right)\left( {{Z_1},{Z_2},...,{Z_r}} \right)

where the right hand side is nothing but

\displaystyle \left( {{\nabla _X}\alpha } \right)\left( {{Z_1},{Z_2},...,{Z_r}} \right) = {\nabla _X}\alpha \left( {{Z_1},{Z_2},...,{Z_r}} \right) - \sum\limits_{k = 1}^n {\alpha \left( {{Z_1},{Z_2},...,{\nabla _X}{Z_k},...,{Z_r}} \right)}.

Note that, the first term of the right hand side of the above identity is covarian derivative acting on an (0,s)-tensor. This action can be defined by the following rule

\displaystyle {\nabla _X}\left( {{Z_1} \otimes {Z_2} \otimes\cdots\otimes {Z_s}} \right) = \sum\limits_{k = 1}^s {{Z_1} \otimes {Z_2} \otimes\cdots\otimes {\nabla _X}{Z_k} \otimes\cdots\otimes {Z_s}}.

We are now in a position to define the second order covariant derivative, denoted by \nabla^2. To be exact, we define

\displaystyle {\nabla ^2}:{C^\infty }({\bigotimes\nolimits^{r,s}}M) \to {C^\infty }({\bigotimes\nolimits^{r + 2,s}}M)

where

\displaystyle\begin{gathered}\left( {{\nabla ^2}\alpha } \right)\left( {X,Y,{Z_1},{Z_2},...,{Z_r}} \right) = {\nabla _X}\left( {\nabla \alpha } \right)\left( {Y,{Z_1},{Z_2},...,{Z_r}} \right) \hfill \\ \qquad= \left[ {{\nabla _X}\left( {\nabla \alpha (Y)} \right) - \nabla \alpha ({\nabla _X}Y)} \right]\left( {Y,{Z_1},{Z_2},...,{Z_r}} \right) \hfill \\ \qquad= {\nabla _X}\left( {{\nabla _Y}\alpha } \right)\left( {{Z_1},{Z_2},...,{Z_r}} \right) - {\nabla _{{\nabla _X}Y}}\alpha \left( {{Z_1},{Z_2},...,{Z_r}} \right). \hfill \\ \end{gathered}

The higher order can be defined similarly. We shall use the following notation

\displaystyle \nabla _{X,Y}^2\alpha \left( {{Z_1},..,{Z_r}} \right) = {\nabla ^2}\alpha \left( {X,Y,{Z_1},...,{Z_r}} \right).

So now the way to understand our definition for second order covariant derivative is the following

\displaystyle\nabla _{X,Y}^2\alpha= {\nabla _X}{\nabla _Y}\alpha- {\nabla _{{\nabla _X}Y}}\alpha.

We also have another notation, we refer the reader to this topic for further discussion. We define

\displaystyle ({\nabla _i}{\nabla _j}\alpha )\left( {{Z_1},{Z_2},...,{Z_r}} \right) = ({\nabla ^2}\alpha )\left( {\frac{\partial }{{\partial {x^i}}},\frac{\partial }{{\partial {x^j}}},{Z_1},{Z_2},...,{Z_r}} \right).

This notation can be defined for higher order covariant derivatives. We are now interested in computing coefficients. Having

\displaystyle\alpha= \alpha _{{j_1} \cdots {j_r}}^{{k_1} \cdots {k_s}}\frac{\partial }{{\partial {x^{{k_1}}}}} \otimes\cdots\otimes \frac{\partial }{{\partial {x^{{k_s}}}}} \otimes d{x^{{j_1}}} \otimes\cdots\otimes d{x^{{j_r}}}

one can see that

\displaystyle {\nabla _i}\alpha _{{j_1} \cdots {j_r}}^{{k_1} \cdots {k_s}}\frac{\partial }{{\partial {x^{{k_1}}}}} \otimes\cdots\otimes \frac{\partial }{{\partial {x^{{k_s}}}}} = ({\nabla _i}\alpha )\left( {\frac{\partial }{{\partial {x^{{j_1}}}}},...,\frac{\partial }{{\partial {x^{{j_r}}}}}} \right).

Examples. We now compute {\nabla _i}{\nabla _j}f. Clearly,

\displaystyle\begin{gathered}{\nabla _i}{\nabla _j}f = ({\nabla ^2}f)\left( {\frac{\partial }{{\partial {x^i}}},\frac{\partial }{{\partial {x^j}}}} \right) \hfill \\ \qquad= {\nabla _i}{\nabla _j}f - {\nabla _{{\nabla _i}\frac{\partial }{{\partial {x^j}}}}}f \hfill \\ \qquad= \frac{{{\partial ^2}f}}{{\partial {x^i}\partial {x^j}}} - {\nabla _{\Gamma _{ij}^k\frac{\partial }{{\partial {x^k}}}}}f \hfill \\ \qquad= \frac{{{\partial ^2}f}}{{\partial {x^i}\partial {x^j}}} - \Gamma _{ij}^k{\nabla _{\frac{\partial }{{\partial {x^k}}}}}f \hfill \\ \qquad= \frac{{{\partial ^2}f}}{{\partial {x^i}\partial {x^j}}} - \Gamma _{ij}^k\frac{{\partial f}}{{\partial {x^k}}}. \hfill \\ \end{gathered}

This is similar to the Hessian of a function discussed here.

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