Currently, the only way that I know how to determine geodesic completeness in practice is to start with a geodesic and apply Hopf-Rinow. Is there any way to infer information about geodesic completeness without solving the geodesic equations?

More specifically, I had a very naive thought about Riemannian metrics on $\Bbb{R} \times \Bbb{R}_{+}$ that could occur "between" the hyperbolic and Euclidean metrics, so I considered a one-parameter ($\kappa$) family of metrics given in coordinates by $g_{ij}=y^{\kappa-2}\delta_{ij}$. With the substitution $\lambda = \frac{\kappa - 2}{2}$, the corresponding geodesics satisfy the following geodesic equations

$$\ddot{x} + 2\dot{x}\dot{y}\frac{\lambda}{y} = 0$$ $$\ddot{y} + \dot{y}^2 \frac{\lambda}{y} - \dot{x}^{2}\frac{\lambda}{y} = 0$$

The cases where $\lambda = 0$ and $-1$ correspond to the Euclidean and hyperbolic metrics, respectively, and so geodesics are well-studied in these cases. I'm not sure how to solve given general $\lambda$, however, and even Wolfram|Alpha was struggling for other particular values of $\lambda$. This leads me to believe that answering questions about geodesics and completeness may not be best approached by explicit computations as before, but perhaps by some more general techniques. Is there a way to determine geodesic completeness from only the geodesic equations above?

Thank you kindly.

  • $\begingroup$ What is the geometric meaning of the background metrics of these geodesic equations? $\endgroup$ – Mikhail Katz Jun 6 '17 at 15:09
  • $\begingroup$ Good question - I'm not sure there's any real geometric meaning as this idea was really more basic and exploratory in nature. I was curious about whether or not any interesting metrics could occur "in between" the Euclidean and hyperbolic cases. Coordinate-wise, I'm looking at $g_{ij} = y^{k-2} \delta_{ij}$, and then the substitution $\lambda = (k-2)/2$ should yield the above geodesic equations. $\endgroup$ – JoeDub Jun 6 '17 at 15:37
  • $\begingroup$ You should mention this description of the metric explicitly in the question. $\endgroup$ – Mikhail Katz Jun 7 '17 at 6:55

If $\lambda\not=0$ then this is still the hyperbolic metric but in a different parametrisation; use $Y^2=y^{2-k}$ and $X=x$. Therefore the metric is complete.


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