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.
- 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.
- $grad(f)=\nabla f\ $ the gradiant of $f:M\to\mathbb R$
Has $\nabla$ something to do with the Levi-Civita connection?
- 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.
- 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.