Volumes of submanifolds with respect to two different riemannian metrics Suppose we have a compact manifold $M$ and two riemannian metrics $g_1,g_2$. For each riemannian metric we can define the riemannian measure and for each compact submanifold $Y\subset X$ we have the induced metrics on $Y$ and we can compute the volume of $Y$ with respect to this metric. Now I know that for each $x\in M$ and $v\in T_x M$ we have that there exist constants $a,b>0$ such that $ag_x^1(v,v)\leq g_x^2(v,v)\leq bg_x^1(v,v)$ , using the fact that $SM$ is compact since $M$ is compact. Now I would like to see that for the volume of a submanifold with respect to each metric we get a similar inequality, i.e, that $aVol_1(Y)\leq Vol_2(Y)\leq bVol_1(Y) $, and for that I belive it suffices to get such an equality for $\sqrt{G^{\alpha}}$ , where this denotes the determinant of the matrix $g_{ij}^{\alpha}=g(\frac{\partial}{\partial x_i^{\alpha}},\frac{\partial}{\partial x_j^{\alpha}})$, for each one of the metrics. Does anyone know if this is possible and if so if how can one prove it ?
(My interest in this comes from the Yomdin's Theorem for the topological entropy of the geodesic flow)
 A: I am afraid that, while things can be reduced to smooth functions, which allows you to use a standard compactness argument, a substantial amount of technology is needed to set things up propertly. If you don't know the tools, it will be rather hard to get through this, I hope the description is still helpful.
Let me recall that the square root of the determinant of the Gram matrix of a Riemannian metric constitutes the local coordinate expression for a well defined geometric object called a density. (These are the objects that can be integrated without choosing an orientation.). Technicall speaking, such densities are (smooth) sections of a line bundle, and a crucial feature of the volume density of a Riemannian metric is that it is nowhere zero. (This is a condition that makes sense for sections of a line bunndle.) Now it is a general fact that given a nowhere vanishing section of $\sigma$ of a line bundle, any other section of that bundle can uniquely be written as $f\sigma$ for a smooth function $f$. If the second section is nowhere vanishing, too, then $f$ is nowhere vanishing. On a compact manifold, you can then conclude that $f$ has a maximum $b$ and a minimum $a$, which have to have the same sign and for volume densities you conclude that they are both positive since both densities lead to positive integrals. But then of course $a\sigma\leq f\sigma\leq b\sigma$ and monotonicity of the integral implies that volume of your manifold with respect to $f\sigma$ lies between $aV$ and $bV$, where $V$ is the volume with respect to $\sigma$. This explains how to apply compactness arguments to $\sqrt{\det(g_{ij})}$.
To obtain what you are looking for, there is the additional problem that you restrict the metric to submanifolds of smaller dimensions, and it is not easy to see what this does to the volume density. (This was the point-wise problem I mentioned in my comment.) I think the way to proceed is as follows: There is a fiber bundle $Gr(k,TM)\to M$ for which the fiber over $x$ is the Grassmannian of $k$-dimensional subspaces of $T_xM$. (This can be easily constructed as an associated bundle to the frame bundle of $M$.) Since the Grassmannian is compact, also the total space of $Gr(k,TM)$ is compact. You an also construct a line bundle $L\to Gr(k,TM)$ (again as an associated bundle) such that smooth sections of $L$ correspond to a (smooth) family of volume densities on a each $k$ dimensional subspace in each tangent space. Now given a Riemannian metric on $M$, restricting $g_x$ to each $k$-dimensional subspace of $T_xM$ and taking the square root of the determinant of a Gram matrix defines a nowhere vanishing section of $L\to Gr(k,TM)$ which is smooth by construction. Doing this for two metrics, you get a smooth function $f$ relating the two nowhere vanishing sections and by compactness you get a maximum and a minimum.
Now given a $k$-dimensional submanifold $Y\subset M$, you get a canonical embedding $Y\to Gr(k,TM)$ by sending each point $y\in Y$ to the subspace $T_yY\subset T_yM$. The volume density of the restriction of a Riemannian metric to $Y$ can then be obtained as a pullback along this embedding of the section of $L\to Gr(k,TM)$ along this embedding. Doing this for two metrics the sections are thus related by the restriction of $f$ to the image of the emedding, so the global minimum and maximum of $f$ provide bounds for the volume density of $Y$. Integrating, you then get the required bounds on the volumes.
