In this paper, without assuming that manifolds are spin, we prove that if a compact orientable, and connected Riemannian manifold
(
M
n
,
g
)
(M^{n},g)
with scalar curvature
R
g
≥
6
R_{g}\geq 6
admits a non-zero degree and
1
1
-Lipschitz map to
(
S
3
×
T
n
−
3
,
g
S
3
+
g
T
n
−
3
)
(\mathbb {S}^{3}\times \mathbb {T}^{n-3},g_{\mathbb {S}^{3}}+g_{\mathbb {T}^{n-3}})
, for
4
≤
n
≤
7
4\leq n\leq 7
, then
(
M
n
,
g
)
(M^{n},g)
is locally isometric to
S
3
×
T
n
−
3
\mathbb {S}^{3}\times \mathbb {T}^{n-3}
. Similar results are established for non-compact cases as
(
S
3
×
R
n
−
3
,
g
S
3
+
g
R
n
−
3
)
(\mathbb {S}^{3}\times \mathbb {R}^{n-3},g_{\mathbb {S}^{3}}+g_{\mathbb {R}^{n-3}})
being model spaces. We observe that the results differ significantly when
n
=
4
n=4
compared to
n
≥
5
n\geq 5
. Our results imply that the
ϵ
\epsilon
-gap length extremality of the standard
S
3
\mathbb {S}^3
is stable under the Riemannian product with
R
m
\mathbb {R}^m
,
1
≤
m
≤
4
1\leq m\leq 4
(see
D
3
D_{3}
. Question in Gromov’s paper [Foundations of mathematics and physics one century after Hilbert, Spring, Cham, 2018, p. 153]).