Let
M
~
\tilde M
denote a complete simply connected manifold of nonpositive sectional curvature. For each point
p
∈
M
~
p \in \tilde M
let
s
p
{s_p}
denote the diffeomorphism of
M
~
\tilde M
that fixes
p
p
and reverses all geodesics through
p
p
. The symmetry diffeomorphism group
G
∗
{G^{\ast }}
generated by all diffeomorphisms
{
s
p
:
p
∈
M
~
}
\{ {s_p}:\,p \in \tilde M\}
extends naturally to group of homeomorphisms of the boundary sphere
M
~
(
∞
)
\tilde M(\infty )
. A subset
X
X
of
M
~
(
∞
)
\tilde M(\infty )
is called involutive if it is invariant under
G
∗
{G^{\ast }}
. Theorem. Let
X
⊆
M
~
(
∞
)
X \subseteq \tilde M(\infty )
be a proper, closed involutive subset. For each point
p
∈
M
~
p \in \tilde M
let
N
(
p
)
N(p)
denote the linear span in
T
p
M
~
{T_p}\tilde M
of those vectors at
p
p
that are tangent to a geodesic
γ
\gamma
whose asymptotic equivalence class
γ
(
∞
)
\gamma (\infty )
belongs to
X
X
. If
N
(
p
)
N(p)
is a proper subspace of
T
p
M
~
{T_p}\tilde M
for some point
p
∈
M
~
p \in \tilde M
, then
M
~
\tilde M
splits as a Riemannian product
M
~
1
×
M
~
2
{\tilde M_1} \times {\tilde M_2}
such that
N
N
is the distribution of
M
~
\tilde M
induced by
M
~
1
{\tilde M_1}
. This result has several applications that include new results as well as great simplifications in the proofs of some known results. In a sequel to this paper it is shown that if
M
~
\tilde M
is irreducible and
M
~
(
∞
)
\tilde M(\infty )
admits a proper, closed involutive subset
X
X
, then
M
~
\tilde M
is isometric to a symmetric space of noncompact type and rank
k
⩾
2
k \geqslant 2
.