Let
e
(
p
)
e(p)
and
G
(
p
)
G(p)
be the unit outer normal and the Gauss-Kronecker curvature of an oriented closed even-dimensional hypersurface
M
M
of dimension
n
n
in
E
n
+
1
{E^{n + 1}}
. Then for a fixed unit vector
c
c
in
E
n
+
1
{E^{n + 1}}
, we have
\[
(
1
)
∫
M
(
c
⋅
e)
m
G
d
V
=
c
n
+
m
χ
(
M
)
/
c
m
,
a
m
p
;
for
m
=
0
,
2
,
4
,
⋯
,
=
0
,
a
m
p
;
for
m
=
1
,
3
,
5
,
⋯
,
(1)\qquad \begin {array}{*{20}{c}} {\int _M^{} {{{({\text {c}}\cdot {\text {e)}}}^m}GdV = {c_{n + m}}\chi (M)/{c_{m,}}} } & {{\text {for}}\;m = 0,2,4, \cdots ,} \\ { = 0,} & {{\text {for}}\;m = 1,3,5, \cdots ,} \\ \end {array}
\]
where
c
⋅
e
{\text {c}} \cdot {\text {e}}
denotes the inner product of
c
c
and
e
e
the area of
m
m
-dimensional unit sphere, and
χ
(
M
)
\chi (M)
the Euler characteristic of
M
M
.