Whats the metric on a 4-dimensional hyperboloid? $\begingroup$ @LeeMosher OP has given the pullback of the Euclidean metric to the hyperboloid in (global) local coordinates on the hyperboloid We assume (perhaps wrongly) that OP is looking for the induced metric on the hyperboloid in $\Bbb R^5$, in terms of coordinates on the hyperboloid $\endgroup$ –
Shortest path on hyperboloid - Mathematics Stack Exchange You might want to look at A Pressley, Differential Geometry, text problem 8 1, he calculates the geodesics for the hyperboloid of one sheet \begin{equation} x^2+y^2-z^2=1 \end{equation} and finds that they are actually four, two straight lines, one circle and one hyperbola
Show that the hyperboloid of one sheet is a doubly ruled surface. Show that the hyperboloid of one sheet is a doubly ruled surface, i e each point on the surface is on two lines lying entirely on the surface (Hint: Write equation (1 35) as $\frac{x^2}{a^2}-\frac{z^2}{c^2} = 1 - \frac{y^2}{b^2}$, factor each side Recall that two planes intersect in a line
calculus - Deriving parameterization for hyperboloid - Mathematics . . . I know there is a parameterization of a hyperboloid $\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1$ in terms of $\cosh$ and $\sinh$, but I don't see how these equations are derived I would appreciate it if either someone could explain to me how such a parameterization is derived or recommend a reference
geometry - Hyperboloid Equation - Mathematics Stack Exchange The result is a hyperboloid of two sheets contained within a double cone The equation $$\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1, \tag2$$ on the other hand, is a hyperboloid of one sheet
differential geometry - Hyperboloid model: isometric embedding of . . . The hyperboloid model of the hyperbolic plane allows one to view the hyperbolic plane as a submanifold of Minkowski space $\mathbb R^{2,1}$, through the parametrisation $$ x_1 = \sinh u\cos v \\ x_2 = \sinh u\sin v \\ x_3 = \cosh u $$ The hyperbolic plane can also be viewed abstractly as the space $\mathbb R^2$ equipped with the Riemannian