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I am struggling to understand the motivation for the definition of the Gram Matrix and its corresponding role for the definition of the Lebesgue measure on submanifolds.

$M \subset \mathbb{R}^n\ k$ dimensional $C^1$-Submanifold

$\phi^{-1}: T \subseteq \mathbb{R}^m \to M$ local parametrization of M

We define the Gram-Matrix as

$G_\phi: T \to \mathbb{R}^{n\times n}$

$t \mapsto (D(\phi^{-1}(t))^TD(\phi^{-1}(t)) $

and its corresponding determinant at $t\in T: g_\phi(t)$.

Using this one defines the m-dimensional-Lebesgue measure $\lambda^m$ on the Submanifold M (for simplification assume that there exists a map which already describes all of $M$, i.e.:

$ \phi: M \mapsto T \subset \mathbb{R}^m $, the Atlas of M is only one map.)

Then one defines:

$\lambda_M: \mathbb{B}^d \cap M \to [0,\infty]$

$ B \mapsto \int_{\phi(B)} (g_{ \phi^{-1}}(t))^{1/2}d\lambda^m(t) $

I think the usage of the Gram-Determinant accounts for some deformation of $\phi$, similar to the Transformation Rule, but the construction of the Gram-Matrix does not make sense to me yet.

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Recall that for linear independent vectors $v_1, \cdots, v_k \subset \mathbb{R}^n$ the $k$-dimensional volume of the k-parallelepiped $P$ is given by $Vol_{k} = \sqrt{\det{A^TA}}$ where $A \subset \mathbb{R}^{n \times k}$ is the matrix containing the column vectors $v_1, \cdots, v_k$ (we assume $k \leq n$).

Let $M \subset \mathbb{R}^n$ be a $k$-dimensional submanifold with one atlas $\Phi: V \to M$. Let's assume we devide $V$ into finitely many small cubes $Q_i$. We might want to write $Vol_k(M) \approx \sum_i Vol_k(\phi(Q_i))$ similarly to Riemann-sums. If we make the cubes "small enough" we have $\phi \approx D\phi$ since the Jacobi Matrix is the liniarization of $phi$.

Now replacing the sum by an integral and $Vol_k(..)$ by $\sqrt{D\phi^T D \phi}$ yields the representation. I hope this gives some motivation on the usage of the gram matrix.

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