5
$\begingroup$

do Carmo gives a definition of sectional curvature as follows:

$K(x,y) = \frac{\langle R(x,y)x,y\rangle}{|x\times y|^2}$

where $x,y \in T_pM$ are linearly independent vectors.

My question: The curvature of a riemannian manifold is a correspondence that associates, for each vector fields $X,Y$, a linear map $R(X,Y)$, which takes vector fields in vector fields. In the definition above $x,y \in T_pM$ which means they are not vector fields, so how could one interpret $R(x,y)x$?

  • 0
    one upvote for being perseverant ;-)2012-06-02

1 Answers 1

5

One of the main points about tensor fields is that they can be evaluated pointwise. That is, you don't need vector fields, but only tangent vectors. This applies to all three arguments of the curvature tensor (your $x, y$).

Edit after reading M.B.s comment: the point about his remark is, that the result will not depend on the choice of vector fields. Too bad, he removed the comment. It said (wording may differ): choose vector fields $X, Y$ such that $X(p)=x, Y(p)= y$ and evaluate along these.

  • 0
    that's right, thanks :)2012-06-02