Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am reeding the book by Aubin on Differential Geometry.

Let $D_XY$ be the covariant derivative of the vector field $Y$ in the direction of the vector $X$.

We know that $$D_XY = X^iD_iY=X^i(\partial_i Y^j)\frac{\partial}{\partial x^j} + X^iY^jD_i\left(\frac{\partial}{\partial x^j}\right)$$ where $D_i = D_{\frac{\partial}{\partial x^i}}$.

Then the author defines: $\nabla Y$ is the differential (1,1)-tensor which in a local chart has $(D_iY)^j$ as components (the $j$th component of the vector field $D_iY$.) The above implies that $\nabla_i Y^j= (D_iY)^j$.

1) This equality follows by definition if I am right. the book states it a bit confusingly.

2) What is the $\nabla$ thing called or mean? i can't find it in any book. What is the use of it?

share|cite|improve this question
up vote 3 down vote accepted
  1. Yes.

  2. As you found out in type-setting your question, the symbol $\nabla$'s name is "nabla". It is one of the myriad symbols used to denote some sort of differentiation/derivation. For some historical notes you can consult Wikipedia. You should think of $\nabla Y$ as a single, compound symbol for expression the tensor field defined, much in the same way $\partial B$ when $B$ is a topological subspace of a larger topological space $X$ is a compound symbol referring to a particular subset of $X$.

Why does Aubin give two different symbols?

I don't know, but I can hazard a guess.

A derivation such as $D_X$ is not, in general, necessarily tensorial in $X$ (in the sense that it is $C^\infty(M)$-linear). For example, consider the Lie derivative $L_X$. That $X \to D_XY$ is tensorial in $X$ is a (somewhat) special property of covariant differentiation. Perhaps Aubin wanted to emphasize this difference pedagogically, and thus introduce a different symbol for "the operation" and for "the end result". This can also be akin to the distinction in multivariable calculus between the partial derivative $\partial_{x^i} f$, and the gradient vector $\nabla f$. The directional derivative of $f$ in the direction $v$ is often written (in multivariable calculus textbooks) as $v\cdot \nabla f$, and not $\partial_v f$.

share|cite|improve this answer
Thanks. Regarding 2, I meant more what is the difference/significance between $D_X$ as defined above and $\nabla_X$ as defined above. In most books that I see, they just define a covariant derivative $D_X$ (but use the symbol $\nabla_X$) and that's that. Why does Aubin give two versions? – hopo2 Sep 12 '12 at 16:02

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.