  Magnetic kagome metals, in which topologically non-trivial band structures
and electronic correlation are intertwined, have recently emerged as an
exciting platform to explore exotic correlated topological phases, that are
usually not found in weakly interacting materials described within the
semi-classical picture of electrons. Here, via a comprehensive single-crystal
neutron diffraction and first-principles density functional theory study of the
archetypical topological kagome metal Mn$_3$Sn, which is also a magnetic Weyl
fermion material and a promising chiral magnet for antiferromagnetic
spintronics, we report the realisation of an emergent spin-density wave (SDW)
order, a hallmark correlated many-body phenomenon, that is engineered by the
Fermi surface nesting of topological flat bands. We further reveal that the
phase transition, from the well-known high-temperature coplanar and
non-collinear k = 0 inverse triangular antiferromagnetic order to a double-$k$
non-coplanar modulated incommensurate magnetic structure below $T_1$ = 280 K,
is primarily driven by the SDW instability. The double-$k$ nature of this
complex low-temperature magnetic order, which can be regarded as an intriguing
superposition of a longitudinal SDW with a modulation wavevector k$_L$ and a
transverse incommensurate helical magnetic order with a modulation wavevector
k$_T$, is unambiguously confirmed by our observation of the inter-modulation
high-order harmonics of the type of 2k$_L$+k$_T$. This discovery not only
solves a long-standing puzzle concerning the nature of the phase transition at
$T_1$, but also provides an extraordinary example on the intrinsic engineering
of correlated many-body phenomena in topological matter. The identified
multi-$k$ magnetic state can be further exploited for the engineering of the
new modes of magnetization and chirality switching in antiferromagnetic
spintronics.
