5
$\begingroup$

I am reading the paper “ The first eigenvalue of a small geodesic ball in a riemannian manifold”, by Karp and Pinsky, from where I took the following:

product

Here, $\Delta_{-2}$ denotes the usual Laplacian in $\mathbb{R}^n$, $R$ is the curvature tensor of the manifold and $\rho$ is defined by

$$ \rho(x) = \sum_{i,j} \rho_{ij} x_i x_j$$

where $\rho_{ij}$ are the components of the Ricci tensor of the manifold.

What does $R\#R$ mean? It should be a kind of product between $(4,0)$ tensors that outputs a function. Have you seen this before?

P.S. the reference they cite is not available online.

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ What's the reference? $\endgroup$
    – Deane Yang
    Commented Mar 17, 2022 at 18:18
  • 2
    $\begingroup$ Usually, what you end up doing is calculating it yourself. Any chance you tried that already? $\endgroup$
    – Deane Yang
    Commented Mar 17, 2022 at 18:19
  • $\begingroup$ The reference is: A. Gray, M. Pinsky. The mean exit time from a small geodesic ball in a riemannian manifold, Bulletin des Sciences Mathématiques, 2eme série 107(1983), 345-370. $\endgroup$
    – Eduardo Longa
    Commented Mar 17, 2022 at 19:47
  • $\begingroup$ a superscript # means "take the traceless part of the tensor" , perhaps the typed text did not allow for a superscript # ? $\endgroup$
    – Carlo Beenakker
    Commented Mar 17, 2022 at 21:11

0

Reset to default

You must log in to answer this question.