Induced Metric for Riemann Hypersurface in Euclidean Signature

[craigthone]

We know in Lorentzian signature spacetime, in the case of timelike or spacelike hypersurfaces $\Sigma$ with
\begin{align}
n^\alpha n_\alpha=\epsilon=\pm1
\end{align}
where $\epsilon=1$ for timelike and $-1$ for spacelike. We can define a tensor $h_{\alpha\beta}$ on $\Sigma$ by
\begin{align}
h_{\alpha\beta}=g_{\alpha\beta}-\epsilon n_\alpha n_\beta
\end{align}
What is the corresponding relation for hypersuface in Euclidean signature manifold.

 

[pervect]

We know in Lorentzian signature spacetime, in the case of timelike or spacelike hypersurfaces $\Sigma$ with
\begin{align}
n^\alpha n_\alpha=\epsilon=\pm1
\end{align}
where $\epsilon=1$ for timelike and $-1$ for spacelike. We can define a tensor $h_{\alpha\beta}$ on $\Sigma$ by
\begin{align}
h_{\alpha\beta}=g_{\alpha\beta}-\epsilon n_\alpha n_\beta
\end{align}
What is the corresponding relation for hypersuface in Euclidean signature manifold.

An example might help. If you have a metric
$$\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$ , the induced metric for a vector in the z direction, i.e. with components <0,0,1> would be
$$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$

So the same formula works, the induced metric $h_{ab}= g_{ab} - n_a n_b=$ works.

However, you can see that the sign convention for a spacelike vector we used is the opposite of the one you specified.

However, that's using a + sign convention for a Euclidean spacelike hypersurface, which is the opposite of the convention you chose for the Lorentzian matrix. I think using the same sign convention you used would be very confusing, but so is switching the sign convention between the two cases.