Spatial curvature in coincident gauge $f(Q)$ cosmology

In this work we study the Friedmann-Lema\^{i}tre-Robertson-Walker cosmologies with arbitrary spatial curvature for the symmetric teleparallel theories of gravity, giving the first presentation of their coincident gauge form. Our approach explicitly starts with the cosmological Killing vectors and constructs the coincident gauge coordinates adapted to these Killing vectors. We then obtain three distinct spatially flat branches and a single spatially curved branch. Contrary to some previous claims, we show that all branches of connection in the covariant approach can be studied in this gauge-fixed formalism, which offers certain conceptual advantages. Interestingly, we find that the flat and negative spatially curved solutions in $f(Q)$ gravity can be seen as equivalent to the metric teleparallel $f(T)$ theories, demonstrating a deeper connection between these theories. This is accomplished by studying the connection equation of motion, which can be interpreted as a consistency condition in the gauge-fixed approach. Finally, we discuss the role of diffeomorphism invariance and local Lorentz invariance in these geometric modifications of gravity.