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?

  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 $\partial_v f$.

  • $\begingroup$ 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? $\endgroup$ – hopo2 Sep 12 '12 at 16:02

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