Affine connections Definition 0.1. An affine connection r on a manifold M is a method of assigning a vector field rX Y to every pair of vector fields X and Y on M, so that: •r is bilinear 1 1 •r is C linear in X, that is, rfX Y = frX Y for any f 2 C M •r is Leibnitzian in Y , that is, rX (fY ) = X(f)Y + frX Y . An affine connection should be thought of as a way to differentiate Y with respect to the vector field X. Notice that the vector (rX Y )p only depends on Xp, but it depends on Y in a neighborhood of p. Suppose we are given local coordinates, inducing vector fields f@ig. We can then 3 k define n smooth functions Γij by the equation X k r@i @j = Γij@k: k 3 k Conversely, given any choice of n smooth functions Γij, we can define a connection using the above formula, and extending to all vector fields by C1-linearity and P P Leibnitz's rule. Thus if X = i xi@i and Y = j yj@j, then 0 0 11 X X X k rX Y = @ xi @@iyk + ΓijyjAA @k: k i j Definition 0.2. A connection r is symmetric if rX Y − rY X = [X; Y ] for all vector fields X and Y . Given a connection r and a function f : M ! R, define the Hessian of f to f be H (X; Y ) = X(Y (f)) − (rX Y )(f). Notice that, at a critical point of f, the term (rX Y )(f) = df(rX Y ) = 0 and thus this agrees with our previous defini- tion. Also notice that Hf is tensorial: obviously Hf (gX; Y ) = gHf (X; Y ), but f also H (X; gY ) = X(gY (f)) − (rX gY )(f) = X(g)Y (f) + gX(Y (f)) − (grX Y + X(g)Y )(f) = gHf (X; Y ). Proposition 0.3. r is symmetric if an only if, for every function f 2 C1M, f f k H (X; Y ) = H (Y; X). In local coordinates, r is symmetric if and only if Γij = k Γji. f f Proof: By definition, H (X; Y ) − H (Y; X) = ([X; Y ] − rX Y + rY X)(f). For the second claim, note that [@i;@j] = 0 since they are coordinate vector fields, and k k r@i @j = r@j @i if and only if Γij = Γij. Definition 0.4. A connection r is compatible with a metric h·; ·i if it is Leibnitzian with respect to the metric, that is, X(hY; Zi) = hrX Y; Zi + hY; rX Zi. Proposition 0.5. A metric admits a unique compatible symmetric connection (which we call the Levi-Civita connection of the metric). Proof: This is Lemma 8.6 in Milnor. Second fundamental form and applications to Morse theory Proposition 0.6. Let M be a manifold with a metric h·; ·i, and let Q ⊆ M be a submanifold. Let r be the Levi-Civita connection, and let rQ be the Levi-Civita 1 2 connection of the restricted metric. Then, for any vector fields X and Y tangent Q to Q, rX Y is the projection of rX Y to TQ. Proof: Let π : TM ! TQ be the projection. Since the Levi-Civita connection is unique, it suffices to show that π ◦r is a symmetric connection which is compatible with h·; ·i|Q. Immediately it is a symmetric connection. To see that it is compatible with h·; ·i|Q, note that for any vector fields U and V on M, so that V is tangent to Q, then hU; V i = hπ(U);V i. Then use the fact that r is compatible with h·; ·i. Definition 0.7. Let M be a manifold with a metric h·; ·i, and let Q be a subman- ?TQ ifold. Define the second fundamental form of Q by II(X; Y ) = (rX Y ) (where V ?TQ denotes the projection of V onto the othogonal complement of TQ). It is a function with inputs X and Y , vector fields tangent to Q, whose output is a vector field orthogonal to Q. Proposition 0.8. II is a symmetric tensor. Proof: II is clearly tensorial in X, therefore symmetry implies tensoriality. But II(X; Y )−II(Y; X) = [X; Y ]?TQ, and [X; Y ] is tangent to Q whenever X and Y are. In some sense, II measures the failure of Q to be flat inside of M. For example, if M = Rk × Rn and Q is the graph of a function f : Rk ! Rn, then we can represent II be nk2 smooth functions. (k2 inputs for dim Q and n-dimensional output for the orthogonal directions. These functions are exactly the second partial derivatives of f. In particular, II = 0 if and only if f is an affine linear function. Let M be a manifold with a metric, and Q ⊆ M a submanifold. Let TMjQ = π−1(Q), where π : TM ! M is the natural projection. Then dπ is surjective everywhere, so it is transverse to every manifold, and thus TMjQ is a submanifold. ∗ Let ν(Q) = f(q; v) 2 TMjQ; v ? TQqg. Since the map h·; ·i : TMjQ ! T Q has surjective differential everywhere, it follows that ν(Q) is a manifold (the pre-image of the zero section in T ∗Q). Intuitively, ν(Q) is the set of all vectors at Q which are tangent to M but orthogonal to Q. Now, suppose that M = RN with the standard metric. There is a natural map e : ν(Q) ! RN defined by e(q; v) = q + v (the \endpoint" map). N 2 Proposition 0.9. Let Q ⊆ R be a submanifold, and let dp : Q ! (0; 1) be the N 2 distance squared function to a point p 2 R n Q. Then q is a critical point of dp if ? and only if p = e(q; v) for some v 2 TQq . Furthermore, q is a degenerate critical point if and only if (q; v) is a critical point of e. PN Lemma 0.10. For @r = i=1(xi − pi)@i being the radial vector field to the point d2 p, H p (X; Y ) = 2hX; Y i + 2hII(X; Y );@ri. Proof: By translation in RN , we can assume that p = 0. Let D : RN ! R be the 2 2 2 function (x1; : : : ; xN ) 7! x1 +:::+xN , so d0 = DjQ. Since we are using the standard metric we see HD(X; Y ) = 2hX; Y i. But by definition H(X; Y ) = X(Y (D)) − Q ?TQ (rX Y )(D), which is equal to X(Y (D)) − (rX Y )(D) − (rX Y ) (D) (simply by splitting rX Y into its tangential and normal coordinates). Now, assuming that d2 X and Y are tangent to Q, we get 2hX; Y i = H 0 (X; Y ) − II(X; Y )(D). And V (D) = 2hV; @ri for any vector V , since rD = 2@r. 3 Proof of Proposition 0.9: Again we can assume p = 0. q is a critical point if and only if TQq is tangent to a sphere centered at the origin. But this happens if and only if the radius is orthogonal to TQq. This proves the first claim. 0 0 Given a vector w 2 T ν(Q)(q;v), we realize w = (q (0); v (0)) for some path (q(t); v(t)) 2 ν(Q), t 2 (−"; ") with q(0) = q; v(0) = v. The chain rule says that 0 v (0) = rq0(0)V , where V is any vector field orthogonal to Q satisfying V (q(t)) = v(t). Let X be any vector field tangent to Q, then hX; V i = 0 everywhere. Therefore 0 0 0 0 = q (0)(hX; V i) = hrq0(0)X; V i + hX; rq0(0)V i =iII(q (0);X); v(0)i + hX; v (0)i. Since p = 0 = e(q(0); v(0)) = q(0) + v(0), we see that v(0) = −@r(q(0)), and therefore 0 0 hX; v (0)i = hII(q (0);X);@ri for every vector X which is tangent to Q. Suppose (q; v) is a critical point of e, which means that there is a w with de(w) = 0. By definition de(w) = q0(0)+v0(0), so 0 = hde(w);Xi = hq0(0);Xi+hv0(0);Xi = 0 0 d2 0 hq (0);Xi + hII(q (0);X);@ri. Thus by Lemma 0.10 we see that H 0 (q (0);X) = 0 2 for every vector X, showing that q is a degenerate critical point of d0. 2 Conversely, suppose that q is a degenerate critical point of d0, so we have a 0 d2 0 vector q (0) satisfying H 0 (q (0);X) = 0 for all vectors X which are tangent to Q. Choose an arbitrary path of vectors ve(t) which is orthogonal to TQq(t), satisfying 0 ? 0 ve(0) = −@r. Then let (ve (t)) be the component of ve (t) which is orthogonal to 0 ? Q, and let v(t) = ve(t) − t(ve (t)) . v(t) is orthogonal to Q, and v(0) = −@r, and 0 0 0 ? 0 additionally we see that v (0) = ve (0) − (ve (0)) so v (0) is tangent to TQq(0). By the above formula for hX; v0(0)i and Lemma 0.10, we see that hX; v0(0)i = 0 0 −hX; q (0)i. Since v (0) is tangent to TQq(0) and X is an arbitrary vector X which 0 0 is tangent to TQq(0), this implies that v (0) = −q (0).