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6-i M. iimg constant, measures the allowed amount of non-constancy in K(V). Notice, that by sealing one can reduce the general case to that of c e= 1 and then the inequality c< K < 1 says that the sectional curvatures of V are strictly pinched between 1 those of the unit sphere and the one of radius c-'S", Rauch conjectured that the best pinching constant in his theorem must be 1/4. This value is motivated by the fact that the complex protective space <£ P", which goes next in roudness after S\ has the sectional curvatures spread over the closed interval [1/4,1] for the natural U(n -j l)-invariant (Fubini-Study) metric onC-P*.

The universal covering of the cylinder is the Eucldean plane tR~ and the ball B(v,r) in V is the image of a Euclidean 2-bail (disk) B in R-. As r becomes greater than the half-length of S1 the map of B to S1 X 1ft becomes non-one-to-one and we can see B wrapping around the cylinder as r grows. One observes a similar picture in an arbitrary V with the so-called exponential map e : TV(V} —> V which sends each vector r € 3"« (V) to the second end of the geodesic segment in V issuing in the direction of T and having length ==[[rj[.

If 7 is complete we thus obtain a map (called the normal exponential map) d;W y, IR-^F for &{iv, e) = dc (w) in the above notation. It is characterized by the geodesic property of d on the lines w X VI and by the initial conditions I d(w, 0)•== w and — d(iv, 0) = for all i« € IK. (In the non-complete case we have such a map on some neighbourhood U cz W X 1R of ff X 0 c F X R J provided W lies in the interior of V). THE SECOND FUNDAMENTAL FOKM OF W IN F. Using rfE one defines the second fundamental form IPV which measures how much IF is curved inside V (exactly as in the Euclidean ease, see § 0), by 1 A 2 de where g* is the metric on W induced from g by the map d, : W—> V.

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