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1 vote

Can the exponential map be used to define geodesics (and hence, generalisations of geodesics)?

Let’s define $\gamma:\mathbb{R}\to M$ as a $\sigma$-geodesic iff for any $a,b,c\in\mathbb{R}$, there are vectors $v,w$ in the tangent space at $\gamma(a)$ with $$\sigma_{\gamma(a)}(v)=\gamma(b)$$ $$\...
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4 votes

Can the exponential map be used to define geodesics (and hence, generalisations of geodesics)?

Exponential map in your definition is closely related to the smooth family of smooth curves smoothly depending on the position such that in every point in every direction there exists precisely one ...
1 vote

trapped geodesics

Let us shoot a geodesic from a random point in a random direction. Note that with probability 0 it is one-sides finite + other-side infinite. However, if you shoot a geodesic from a nonregular point, ...

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