Abstract
With the aid of concrete examples, we consider the question of whether, in the presence of conformal curvature, a conformal geodesic can become trapped in smaller and smaller sets, or phrased informally: are spirals possible? We do not arrive at a definitive answer, but we are able to find situations where this behaviour is ruled out, including a reduction of the conformal-geodesic equations to quadratures in a specific non-conformally flat metric.
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