Timeline for Tubular neighbourhood of a smooth curve
Current License: CC BY-SA 2.5
6 events
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| Dec 25, 2010 at 16:46 | comment | added | Dick Palais | @Pratik. Sure, a tubular neighborhood always exists, but you asked how large it could be, or rather you asked if there was always a tubular neighborhood of a radius equal to the reciprocal of the minimal geodesic curvature, and my example shows that cannot be the case. | |
| Dec 25, 2010 at 10:10 | comment | added | Pratik | Even though $k_g = 0$, there exists tubular neighbourhood with some $\varepsilon>0$. The problem faced by taking the value of $\varepsilon$ to be half of the circumference is due to the fact that the $exp$ map is not diffeomorphic for that value of $\varepsilon$. Infact the $i(Cylinder)$, injectivity radius of the cylinder is exactly half of the circumference. In any case the tubular neighbourhood must exists if the curve is not self intersecting. | |
| Dec 25, 2010 at 10:04 | vote | accept | Pratik | ||
| Dec 25, 2010 at 10:04 | |||||
| Dec 25, 2010 at 7:30 | history | edited | Dick Palais | CC BY-SA 2.5 |
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| Dec 25, 2010 at 7:15 | history | edited | Dick Palais | CC BY-SA 2.5 |
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| Dec 25, 2010 at 5:34 | history | answered | Dick Palais | CC BY-SA 2.5 |