  Riemann normal coordinates (RNC) at a regular event $p_0$ of a spacetime
manifold $\mathcal{M}$ are constructed by imposing: (i)
$g_{\textsf{ab}}|_{p_0}=\eta_{ab}$, and (ii)
$\Gamma^\textsf{a}_{\phantom{\textsf a}\textsf{bc}}|_{p_0}=0$. There is,
however, a third, $independent$, assumption in the definition of RNC which
essentially fixes the $density$ $of$ $geodesics$ emanating from $p_0$ to its
value in flat spacetime, viz.: (iii) the tangent space
$\mathcal{T}_{p_0}(\mathcal{M})$ is $flat$. We relax (iii) and obtain the
normal coordinates, along with the metric $g_{\textsf{ab}}$, when
$\mathcal{T}_{p_0}(\mathcal{M})$ is a maximally symmetric manifold
$\widetilde{\mathcal M}_{\Lambda}$ with curvature length $|\Lambda|^{-1/2}$. In
general, the "rest" frame defined by these coordinates is non-inertial with an
additional acceleration $\boldsymbol a = - ({\Lambda}/3) \, \boldsymbol x$
depending on the curvature of tangent space. Our geometric set-up provides a
convenient probe of local physics in a universe with a cosmological constant
$\Lambda$, now embedded into the local structure of spacetime as a fundamental
constant associated with a curved tangent space. We discuss classical and
quantum implications of the same.
