# 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.

1. What about general nth-order covariant derivatives?

Comment by G.S. — March 12, 2014 @ 9:23

• Hi, thanks for your interest in my post.

The higher order covariant derivatives can be defined recursively, i.e.

$\displaystyle \nabla^m = \nabla \circ \nabla^{m-1}$

for any $m \geqslant 3$. To be exact, we use

$\displaystyle \nabla^m \alpha (X_1,...X_m,Z_1,...,Z_r)=\nabla_{X_1} (\nabla^{m-1} \alpha) (X_2,...,X_m,Z_1,...,Z_r).$

Comment by Ngô Quốc Anh — March 12, 2014 @ 11:00