Let
M be a real hypersurface of class
C2 in
Cn,
n ≥ 2, and let δ
M(
z) be the Euclidean distance from
z ∈
Cn to
M. In this paper we give an explicit representation in complex tangential direction for the Levi form of the function δ
M (or -log δ
M) by Hermitian and symmetric matrices determined by a local defining function of
M. As its application, we also show that, if
M is defined by a
C2-function
&rho with
d&rho ≠ 0, then the function -log δ
M is strictly plurisubharmonic in complex tangential direction near
M if and only if
M is Levi-flat and the symmetric matrix (δ
2&rho/δ
zi δ
zj ) of degree
n has maximal rank
n - 1 on the complex tangent subspace T
z1,0 (
M) ⊂ T
z1,0 (
Cn) for each
z ∈
M as a linear map. Moreover, we can get directly the well-known Levi condition as the condition in order that -log δ
M is weakly plurisubharmonic in complex tangential direction near and in one side of
M.
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