In most definitions of the geodesic, it is required to be a regular curve,i.e. a smooth curve satisfying that the tangent vector along the curve is not 0 everywhere. I don't know why.
why a geodesic is a regular curve?
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differential-geometry
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1if you allow corners, you can automatically create a shorter segment by taking a short cut across the corner. Draw some pictures. – 2012-12-15