Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Expressing the curvature tensor and connection of a given metric in
  terms of those of another metric   Let $(M,g)$ be a Riemannian manifold, and $m$ be a second metric on $M$. We
give expressions of $m$'s associated connection, and Riemann curvature tensor
$R_m$, in terms of $R_g$ and certain combinations of covariant derivatives of
$m$ (with respect to the Levi-Civita connection associated with $g$). The
formulas turn out to be generalizations of the coordinate expressions.
Coordinate expression formulas can be recovered from ours by setting $g$ as the
Euclidean metric induced by a given coordinate chart. As the covariant
derivative induced by $g$ becomes the ordinary partial derivative and the $R_g$
tensor vanishes, the formulas coincide with the well-known coordinate
expressions for $m$'s connection and curvature tensor.

11
The categories of this paper are: hep-lat
0
The categories of this paper are: hep-ph
33
The categories of this paper are: plasm-ph
7
The categories of this paper are: hep-th
13
The categories of this paper are: quant-ph
21
The categories of this paper are: adap-org
26
The categories of this paper are: bayes-an
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs
25
The categories of this paper are: patt-sol
31
The categories of this paper are: ao-sci
8
The categories of this paper are: hep-ex
19
The categories of this paper are: eess
37
The categories of this paper are: supr-con
23
The categories of this paper are: solv-int
9
The categories of this paper are: nlin