Assume I have a manifold $M$ with $dim(M) = m$. The manifold is equipped with a coordinate chart $x_{i}$ such that the given metric on the manifold is:
$g_{ij} = \delta_{ij} + \frac{x_{i}x_{j}}{1-\sum_{k}x^{2}_{k}}$ and the coordinates satisfy $\sum_{k}x^{2}_{k}\leq1$.
The manifold has a finite volume that can be calculated in the chart as:
$V = \int_{\sum_{k}x^{2}_{k}\leq1} \sqrt{|g|}dx$
Assume I have a submanifold $N$ embedded in $M$ which is defined by the parameterization:
$x_{i} = \sum_{j}A_{ij}\theta_{j}$ with $A\in \mathbb{R}^{[mXn]}$ and $n<m$ is the dimension of the submanifold
As far as I understand, the induced metric on the submanifold is:
$\tilde{g}_{ij}=\sum_{kl}A^{T}_{ik}g_{kl}A_{lj}$
and the volume of the submanifold is
$\tilde{V} = \int_{\sum_{k}x^{2}_{k}\leq1} \sqrt{|\tilde{g}|}d\theta$
My intuition tells me that since the submanifold is embedded in the original manifold its volume should be in some sense smaller than the original manifolds volume (i.e $\tilde{V}<V$). Numerically evaluating the integrals for a specific $A$ I get that it is wrong in general. I'm trying to figure out wether there is a problem with my code or math and I suspect that my problem is with the math side of things, so I have two questions:
1.Is it true that $\tilde{V}<V$ in the general case or even in my specific case? I understand that in some sense I am comparing volumes of different dimensions so it might be nonsense.
2.Is there a sense in which I can get an $m$ dimensional volume characterizing the submanifold $N$, which is comparable to $V$ and smaller than $V$, in the sense that "we are only counting part of the manifold"?
Something like $\int_{U}\sqrt{|g|}dx \ \ \ U=\{x|\exists \theta \ \ s.t \ \ x=A\theta \}$. Is it possible to represent this integral as integration over the parameters $\theta$?
if so I'd be glad to get a reference for the relevant information.