Let
p
E
→
B
p\:E\to B
be a manifold approximate fibration between closed manifolds, where
dim
(
E
)
≥
4
\dim (E)\ge 4
, and let
M
(
p
)
M(p)
be the mapping cylinder of
p
p
. In this paper it is shown that if
g
B
×
I
→
B
×
I
g\: B\times I\to B\times I
is any concordance on
B
B
, then there exists a concordance
G
M
(
p
)
×
I
→
M
(
p
)
×
I
G\:M(p)\times I \to M(p)\times I
such that
G
|
B
×
I
=
g
G|B\times I=g
and
G
|
E
×
{
0
}
×
I
=
i
d
E
×
I
G|E\times \{0\}\times I= id_{E\times I}
. As an application, if
N
n
N^n
and
M
n
+
j
M^{n+j}
are closed manifolds where
N
N
is a locally flat submanifold of
M
M
and
n
≥
5
n\ge 5
and
j
≥
1
j\ge 1
, then a concordance
g
N
×
I
→
N
×
I
g\: N\times I\to N\times I
extends to a concordance
G
M
×
I
→
M
×
I
G\:M\times I\to M\times I
on
M
M
such that
G
|
N
×
I
=
g
G|N\times I=g
. This uses the fact that under these hypotheses there exists a manifold approximate fibration
p
E
→
N
p\: E\to N
, where
E
E
is a closed
(
n
+
j
−
1
)
(n+j-1)
-manifold, such that the mapping cylinder
M
(
p
)
M(p)
is homeomorphic to a closed neighborhood of
N
N
in
M
M
by a homeomorphism which is the identity on
N
N
.