In the paper of Bianchi-Egnell https://scholar.google.com/scholar_url?url=https://www.sciencedirect.com/science/article/pii/002212369190099Q/pdf%3Fmd5%3Dd48dc0fcf8ea636d1d3e06c618ebee8c%26pid%3D1-s2.0-002212369190099Q-main.pdf&hl=en&sa=T&oi=ucasa&ct=ufr&ei=X07iY4GUI86vywTMpJy4AQ&scisig=AAGBfm2Z5MaB9iWvbXiQh5eLzXySF7daVw

They have showed the following discrepancy result for the classical Sobolev embedding i.e $||\nabla\phi||_{2}^2-S^2||\phi||_{2^*}^2\geq\alpha d(\phi,M)^2$. This is in Euclidean set up. Are there analogous result for known manifolds like on Spheres or hyperbolic manifolds if anyone can suggest some references where this kind of results have been shown in some manifolds other than $\mathbb{R^n}$.

In the paper [1] by Bianchi and Egnell, they proved the following discrepancy result for the classical Sobolev embedding $$ \|\nabla\phi\|_{2}^2-S^2\|\phi\|_{2^*}^2\geq\alpha d(\phi,M)^2. $$ This result holds in an Euclidean set up. Are there analogous results for standard manifolds like spheres or hyperbolic manifolds?
Can anyone suggest some references where this kind of results have been shown in manifolds other than $\Bbb R^n$?

Reference

[1] Gabriele Bianchi, Henrik Egnell, "A note on the Sobolev inequality", (English) Journal of Functional Analysis 100, No. 1, 18-24 (1991), MR1124290, Zbl 0755.46014.