Regge-Teitelboim cosmology driven by a curved background

We derive the dynamic equations governing a Regge Teitelboim (RT) brane that represents our universe considering that its degrees of freedom are the embedding functions, evolving in a five dimensional curved spacetime. Within this framework, we investigate the effects of embedding a Friedmann Robertson Walker (FRW) metric into a specific curved ambient spacetime. This approach allows us to analyze in detail how the ambient spacetime's curvature influences the brane universe's dynamics. Specifically, we examine the relationship between the Hubble parameter $H$ and the redshift $z$, demonstrating that our results significantly agree with actual observational data. This is contrasted and compared with other approaches.