By Michael Spivak

Publication by way of Michael Spivak, Spivak, Michael

**Read or Download A Comprehensive Introduction to Differential Geometry, Vol. 1, 3rd Edition PDF**

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**Additional resources for A Comprehensive Introduction to Differential Geometry, Vol. 1, 3rd Edition**

**Example text**

The one we imitate here is Smale’s proof. 5. Since the critical values of f are distinct, let us arrange them in order: Crit(f ) = {c1 , . . , cq } with f (c1 ) > f (c2 ) > · · · > f (cq ), and let αi = f (ci ). The proof of the theorem will use an induction based on the following lemma. 8. Let j ∈ {1, . . , q} and let ε > 0. There exists a good approximation X (in the C1 sense) of X such that: (1) The vector ﬁeld X coincides with X on the complement of f −1 ([αj + ε, αj + 2ε]) in V . (2) The stable manifold of cj (for X ) is transversal to the unstable manifolds of all critical points, that is, s WX (cj ) u WX (ci ).

12 shows the gradient lines that connect the two critical points (here we use the gradient for the “ﬂat” metric on the torus, that is, for the usual metric on R2 ; as we can see, the lines in question form true right angles with the level sets). We can clearly see that W s (a) is the open square (that is, an open disk), that W s (b) is an open interval (the horizontal side of the square in the ﬁgure), as are W u (b) (vertical segment), W s (c) (vertical side) and W u (c) (horizontal segment), while W s (d) is reduced to d.

0]), we attach a disk of (real) dimension 2 (we now have P1 (C), the cell of points [a, b, 0, . . , 0]), and so on. 11. We begin with the minimum (this is precisely our point [1, 0, . . , 0]), where the function has value 0. The ﬁrst critical value is then 1. This corresponds to the critical point [0, 1, 0, . . , 0], which has index 2. . and enables us to attach a cell of dimension 2, and so on. 11. 5. The cell Dk is the piece of the unstable 1 There is also a precise mathematical deﬁnition of the expression “perfect Morse function”, of which the one considered here is the prototype.