# Download PDF by Chern S.S., Li P., Cheng S.Y., Tian G. (eds.): A Mathematician and His Mathematical Work: Selected Papers

By Chern S.S., Li P., Cheng S.Y., Tian G. (eds.)

ISBN-10: 9810223854

ISBN-13: 9789810223854

Those chosen papers of S.S. Chern talk about issues corresponding to essential geometry in Klein areas, a theorem on orientable surfaces in 4-dimensional house, and transgression in linked bundles

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In order to have a more intuitive outlook, let us imagine that the interior diameter of T is very big (so that the "interior" boundary of it represents a high cylinder) in comparison with the exterior one. Thus, we have a deep hole surrounded by the boundary of the full torus, see fig. 7. Let us consider the ball D 3 and cut a circle from the plane, as shown in Fig. a. Let us rotate the part of this cut (that is a circle with two marked points) in the direction indicated by arrows. This operation is possible since the full torus T is deleted from D 3 .

However, in order to simplify the situation we shall not restrict ourselves only to this case, and require that Va E r : a 0 a = a. Analogously, the invariance under [12 requires the left invertibility of the operation 0: for any a and b from r, the equation 0 a = b should have only the solution E r (in the case of the three-colour palette, the inverse operation for 0 and the operation 0 itself coincide). Finally, the invariance under [13 implies self of the operation o b) 0 c ~ (a 0 0 (b 0 c) holds.

Show that if both numbers are equal to zero then the knot is trivial. 1. In the sequel, we shall consider only those knots for which at least one of these numbers is not equal to zero. The following fact is left for the reader as an exercise. 3. For a non-self-intersecting curve these numbers are coprime. 3. Trefoil on the torus So, each torus knot passes p times the longitude of the torus, and q times its meridian, where GCD(p, q) = 1. It is easy to see that for any coprime p and q such a curve exists: one can just take the geodesic line {qq\ - p,p ~ O( mod 21r Let us denote this torus knot by T(p, q).