  Manganites are famous mostly for the colossal magnetoresistive effect, which
involves the phase separation between ferromagnetic phase and charge-ordered
CE-type antiferromagnetic phases. Furthermore, manganites contain some typical
magnetic ferroelectrics, e.g. E-type antiferromagnetic $o$-HoMnO$_3$. Here we
re-examined these zigzag-winding antiferromagnetic phases (CE-type and E-type
antiferromagnets) from the topological perspective. Our theoretical analysis
proved that the E-type phase is a weak topological insulator belonging to the
$\mathbb{Z}$ class. In momentum space, we classify the symmetries of this
phase, and find the three symmetry operators for the chiral, particle-hole, and
time-reversal symmetry. The CE-type phase can be described by the
Duffin-Kemmer-Petiau algebra, implying that it is a new class of topological
insulator and hence extends the existing classification. The corresponding
topological end states are demonstrated via numerical calculations, which may
implicate the experimental observed ferromagnetic edge states in manganite
strips (Nat. Commun. 6, 6179 (2015)) and may play a crucial role in the
colossal magnetoresistive effect.
