Let
M
M
be a Riemannian manifold and let
E
→
M
E\to M
be a Hermitian vector bundle with a Hermitian covariant derivative
∇
\nabla
. Furthermore, let
H
(
0
)
H(0)
denote the Friedrichs extension of
∇
∗
∇
/
2
\nabla ^*\nabla /2
and let
V
:
M
→
E
n
d
(
E
)
V:M\to \mathrm {End}(E)
be a potential. We prove that if
V
V
has a decomposition of the form
V
=
V
1
−
V
2
V=V_1-V_2
with
V
j
≥
0
V_j\geq 0
,
V
1
V_1
locally integrable and
|
V
2
|
\left | V_2 \right |
in the Kato class of
M
M
, then one can define the form sum
H
(
V
)
:=
H
(
0
)
∔
V
H(V):=H(0)\dotplus V
in
Γ
L
2
(
M
,
E
)
\Gamma _{\mathsf {L}^2}(M,E)
without any further assumptions on
M
M
. Applications to quantum physics are discussed.