Let $\Sigma_r$ be a topological sphere in a 3-dimensional asymptotically flat Riemannian manifold $M$ with metric $g$, $\{\frac{\partial}{\partial x^i}\}, 1\leq i\leq3$ is the standard coordinate frame in $\mathbb{R}^3$, If the unit normal $v=\sum\limits_{i=1}^3v^i\frac{\partial}{\partial x^i}$, can we define the induced metric $h_{ij}dx^idx^j$ on $\Sigma_r$ in $(M,g)$ with $h_{ij}=g_{ij}-v_iv_j, 1\leq i,j\leq3$? I am totally confused by this expression. Usually the induced metric of 2-dimensional surface would be a $2\times2$ matrix, denoted by $(B_{ij})$ and the area element has the expression $\sqrt{|B_{ij}|}$dudv. If yes, the induced metric here $(h_{ij})$ has rank 2 and the determinant would be 0. What the area element would be like?
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The induced metric is just the restriction of $g$ to the tangent space of the submanifold. Or the restriction of $h$. The $h$, if you raise one index, is the projection operator. – MBN Apr 04 '14 at 12:38