The book "Riemannian Geoemetry" by Peter Petersen says the following on pg 171:
Now recall that linear isometris $L:\Bbb{R}^k\to \Bbb{R}^k$ with $\text{det }L=(-1)^{k+1}$ has $1$ as an eigenvalue.
I've never heard of this theorem. Is this an elementary linear algebra proposition that I'm somehow unable to recall?
On
Yes, this is correct. A linear isometry is also called an orthogonal transformation, preserving orthogonality and length. If $v$ is an eigenvector with eigenvalue $\lambda$ then $\|v\|=\|Lv\|=|\lambda|\|v\| $ and $|\lambda|=1$.The determinant is the product of the eigenvalues.In $R^k$ you have $k$ eigenvalues (some of them may be pairs of complex conjugate ones). A product like that is equal to $(-1)^{k+1}$ only if at least one of the eigenvalues is 1. Say, in $R^3$ an orthogonal transformation with determinant $=1$ has to be a rotation around an axis or the identity (well, this is also a rotation).
Remember, determinant measures the distortion a linear transformation imparts on a space. If the determinant has absolute value one then volumes are preserved.