Let $f:M\to (N,g)$ be an immersion of a compact smooth manifold $M$ into a compact Riemannian manifold $(N,g)$. Let $$ \mathrm{Vol}(M,f^*g)=\int_M d\mathrm{vol}_{f^*g} $$ be the volume of $M$ with respect to the pullback metric $f^*g$.
My question: Can we get an upper bound on the volume of $(M,f^*g)$ in terms of curvature properties of the metric $g$?
One possible idea I had: suppose that $(N,g)$ has Ricci curvature bounded below by $\kappa$, i.e. there exists a constant $\kappa\in \mathbb{R}$ such that $$ \mathrm{Ric}^N\geq (n-1)\kappa. $$ Does it follow that $(M,f^*g)$ also has Ricci curvature bounded below by some $\kappa'\in \mathbb{R}$? If so, then we can bound $\mathrm{Vol}(M,f^*g)$ above using the Bishop-Gromov inequality.