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I can't wrap my head around notation in differential geometry especially the abundant versions of derivation.

Peter Petersen: Riemannian Geometry defines a lot of notation to be equal but I don't really know when one tends to use which version and how to memorize the definitions and properties/identities.

  1. Directional derivative or equivalently the action of a vector field $X$ on a function ($f:M\to\mathbb R$): $X\cdot f=D_Xf=df\cdot X\ $, which is also denoted as $L_Xf$

This is mostly clear except why the notation $D_Xf\ $ exists.

  1. $grad(f)=\nabla f\ $ the gradiant of $f:M\to\mathbb R$

Has $\nabla$ something to do with the Levi-Civita connection?

  1. Lie derivative of vector fields: $L_XY:=[X,Y]= X\cdot Y - X\cdot Y\ $, where the action of one vector field on one another is given by: $X\cdot Y:=D_XY\ $ the directional derivative of $Y$ along an integral curve of the vector field $X$.

Also mostly clear.

  1. The covariant derivative or Levi-Civita connection $\nabla_XY$

Here my understanding stops and my brain starts dripping out of my ears… Are there mnemonics or other ways to get into all those ways of thinking about differentiating on manifolds. And why do most books use coordinates - are they necessary I rather like not using $X=\sum_ia^i\partial_i$ for vector fields especially if the author (ab)uses Einstein sum convention.

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I recommend John A. Thorpe, Elementary Topics in Differential Geometry. In the abstract setting without a metric, the directional derivative $X(f)$ is defined, also the 1-form $df$ given by $df(X) = X(f).$ However, without the ability to takes duals, there is no natural definition of $\nabla f.$ And no natural $\nabla_X Y.$ These can all be done extrinsically as in Thorpe's book, but change if a second, non-isometric, embedding of the abstract manifold is used. It was an observation of Gauss, Theorem Excitingum, that less structure of $\mathbb R^n$ was involved than expected. –  Will Jagy May 27 '12 at 20:47
    
Anyway, in $\mathbb R^n,$ with hypersurface, if $X$ is the vector field given as the velocity vector of a curve that stays within the submanifold, and $f,Y$ are a function and vector field smooth and defined on an open set around the submanifold, then $\nabla f$ and $\nabla_X Y$ are the orthogonal projections onto the tangent hyperplane of th gradient of $f$ and the very ordinary extrisic $\nabla_X Y,$ just taking $Y$ to be $n$ functions. Both behave badly under non-isometries. –  Will Jagy May 27 '12 at 21:17

1 Answer 1

up vote 3 down vote accepted

It's not so complicated. Let's do things systematically.

a) The pure manifold case
On a differentiable manifold $M$ a vector field $X$ is the datum of a smoothly varying tangent vector $X(m)\in T_mM$ at each point $m\in M$.
Given a smooth function $f\in C^\infty(M)$ you obtain a differential form $ {d}f$ , that is a linear form $df(m)\in T^*_m(M)=(T_m(M))^*$ at each $m\in M$ , smoothly varying with $m$. It is given by the formula $(df(m))(v)=v(f)$ for $v\in T_m(M)$.
The smooth function $X(f)=L_X(f)=D_X(f)\in C^\infty(M)$ is then the function $M\to \mathbb R:m\mapsto (df(m))(X(m))$
Let me insist that this does not require any riemannian structure.

b) The riemannian case
If $M,g$ is a riemannian manifold, each vector space $T_x(M)$ has a euclidean structure, which permits us to associate to each linear form $\phi\in T_m^*(M)$ the vector $v\in T_m(M)$ such that $g_x(v,w)=\phi (w)$ for all $w\in T_m(M)$.
If $\phi=df(m)$ the corresponding $v$ is denoted by $grad(f)(m)$.
This gives rise to the required function $grad(f)\in C^\infty(M).$

c) The Lie derivative
Given two vector fields $X,Y$ on $M$ you can associate to them the following map $$z:C^\infty(M)\to C^\infty(M): f\mapsto X(Y(f))-Y(X(f))$$ A fundamental result is then that to this map corresponds a unique vector field $Z$ such that $z(f)=Z(f)$ for all $f\in C^\infty(M)$.
We then write $Z=[X,Y]$ or $Z=L_X(Y)$
Again, this does not require a riemannian structure on $M$.

(It is better at this stage not to mention connections, which are supplementary data on vector bundles, related to differential operators. If you like, you can ask another question about them.)

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