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Consider the surface in $\mathbb R^{n+1}$ $$H=\left\{(x^1,\cdots,x^{n+1})\in\mathbb R^{n+1}\mid\sum_{i=1}^n(x^i)^2-(x^{n+1})^2=-1,x^{n+1}>0\right\}$$ Prove that the tensor field

$$h=\sum_{i=1}^n\mathrm dx^i\otimes\mathrm dx^i-\mathrm dx^{n+1}\otimes\mathrm dx^{n+1}$$ restricted to $H$ is a Riemann metric, and compute the sectional curvature.

I tried to use the coordinate $(u_1,\cdots,u_n,\sqrt{1+u_1^2+\cdots+u_n^2})$ to prove $h\vert_H$ is positive definite, but it is too difficult to compute curvature. Please help me.

user8891548
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    It is Minkowski metric on hyperboloid model. How about you calculate the Guassian curvature for $n=2$ first? Then try higer dimension by calculating the shape operator $S\partial_i=-\nabla_{\partial_i}N$, you can also find something useful at page 38 (hyperboloid model) and page 148 (calculating curvature) in Riemannian manifold: An introduction to curvature by John M. Lee – H-H Jun 26 '18 at 17:14
  • Thanks a lot, the material you gave are very helpful. – user8891548 Jun 27 '18 at 01:41

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