  Correlations between the elliptic or triangular flow coefficients $v_m$
($m$=2 or 3) and other flow harmonics $v_n$ ($n$=2 to 5) are measured using
$\sqrt{s_{NN}}=2.76$ TeV Pb+Pb collision data collected in 2010 by the ATLAS
experiment at the LHC, corresponding to an integrated lumonisity of 7
$\mu$b$^{-1}$. The $v_m$-$v_n$ correlations are measured in midrapidity as a
function of centrality, and, for events within the same centrality interval, as
a function of event ellipticity or triangularity defined in a forward rapidity
region. For events within the same centrality interval, $v_3$ is found to be
anticorrelated with $v_2$ and this anticorrelation is consistent with similar
anticorrelations between the corresponding eccentricities $\epsilon_2$ and
$\epsilon_3$. On the other hand, it is observed that $v_4$ increases strongly
with $v_2$, and $v_5$ increases strongly with both $v_2$ and $v_3$. The trend
and strength of the $v_m$-$v_n$ correlations for $n$=4 and 5 are found to
disagree with $\epsilon_m$-$\epsilon_n$ correlations predicted by
initial-geometry models. Instead, these correlations are found to be consistent
with the combined effects of a linear contribution to $v_n$ and a nonlinear
term that is a function of $v_2^2$ or of $v_2v_3$, as predicted by hydrodynamic
models. A simple two-component fit is used to separate these two contributions.
The extracted linear and nonlinear contributions to $v_4$ and $v_5$ are found
to be consistent with previously measured event-plane correlations.
