Kurven Auf Orientierbaren Flächen

Let T denote a closed oriented surface and let there be given a basis ₁,..., _n, ₁,..., _n of H₁ (T; ) with _i · _j = _i · _j = 0, _i · _j = _ij as intersection numbers. Then one can construct an ordinary imbedding of T in 3-dimensional euklidian space, such that the given basis is represented by the meridians and parallels of latitude of that imbedding. If there is a given imbedding of T into an euclidian space of dimension n5, then one has a factorisation through an ordinary imbedding into 3-dimensional space, such that the given basis is represented by the meridians and parallels of latitude of that ordinary imbedding. If in the case n=4 the imbedding of T can be factorised through 3-dimensional space one has a further invariant. Besides the skewsymmetric intersection form there is a quadratic form which must take its normal form on the given basis in order to represent this basis by meridians and parallels of latitude as above.