Today we talk about high order covariant derivative. Recall from this topic that covariant derivative acts on a -tensor, the result is an -tensor. For convenience, we mean

.

Thus

.

Precisely, for an -tensor , one defines as the following

where the right hand side is nothing but

.

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

.

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

where

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

.

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

.

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

.

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

one can see that

.

**Examples**. We now compute . Clearly,

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

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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.

for any . To be exact, we use

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