These notes contain an elementary approach to comparsion theorems and related techniques in Riemannian geometry.A preliminary exposition of fundamental concepts needed in this essentially geometrical approach is given in the first three chapters;finally we describe,with some limitations,the results given by Allamingeon (2),Klingemberg (16)and Warner (30).We tried to show is as simple a way as it was possible some relations between the "first derivative"(differential topology) and the "second derivative"(differential geometry).

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