Analysis of manifolds by Munkres J.R.

By Munkres J.R.

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A compact k-dimensional submanifold of Euclidean space W1 with n > 2k + 1 admits a one-to-one projection to a generic hyperplane. 3. Suppose that for /c-dimensional submanifolds of W1 the theorem is already proved. Then for fc-dimensional submanifolds of E n + 1 it follows immediately from the above lemma and the induction hypothesis. 3. Let k be the submanifold in question. Any oriented line in R defines a direction; that is, all parallel lines with the same orientation have the same direction.

Let L be a nonsingular linear operator that is k-contracting with coefficient q < q(k), and Q a unit ball. Then there exists a covering U(L) of the ellipsoid E = 2L(Q) by balls of radius no greater than qxlk\fk < 1/2, having k-dimensional volume no greater than 1/2: Vk(Uo) < 1/2. Here the constant q(k) may be taken equal to (k + l ) - * / ^ - * - 1 . 2. 38 PROOF. 4. Let L be a nonsingular k-contracting linear operator with coefficient q. Let the ellipsoid LQ have the semiaxes a\ ^ • • • ^ an. Then ai--ak ^q.

In all three cases the corresponding cycles may have homoclinic orbits, but only in the first case does the vector field belong to the boundary of the MorseSmale set. This effect is briefly explained in §4 and in more detail in Chapter 2. Here we describe the local bifurcation of saddlenode cycles. §3. 12. A nonhyperbolic cycle of a vector field is of saddlenode type if exactly one of its multipliers is equal to 4-1, while the others are hyperbolic (not on the unit circle). 4 for the case of maps rather than vector fields.

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