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THE RIEMANNIAN CONNECTION 1. Linear Connections on Tensor Fields

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THE RIEMANNIAN CONNECTION 1. Linear Connections on Tensor Fields LECTURE 5: THE RIEMANNIAN CONNECTION 1. Linear connections on tensor fields Now let M be a smooth manifold, and r a linear connection on (vector fields of) M. We will extend r to a linear connection on all tensor fields. This is very easy for (0; 0)-tensor fields (= functions), since we already have a nice one, 1 1 r : Γ(TM) × C (M) ! C (M); (X; f) 7! rX f := Xf = df(X); which obviously satisfies the two conditions in the definition of linear connections. According to the remark in the previous lecture, a linear connection on (r; s)- tensor fields is a bi-linear map r;s r;s r : Γ(TM) × Γ(⊗ TM) ! Γ(⊗ TM); (X; T ) 7! rX T; that satisfies (1) rfX T = frX T , (2) rX (fT ) = frX T + (Xf)T . Again there are too much choices of linear connections in general, and most of them are not interesting. However, if we impose extra assumptions that all these connections r on (r; s)-tensor fields are related in the following natural way: (3) r coincide with the given connections on Γ(TM) and C1(M), (4) rX (T1 ⊗ T2) = (rX T1) ⊗ T2 + T1 ⊗ rX T2, (5) C(rX T ) = rX C(T ), where C : Γ(⊗r;sTM) ! Γ(⊗r−1;s−1TM) is the contraction map that pairs the first vector with the first covector. Then one can prove Theorem 1.1. Given any linear connection r (on vector fields), there is a unique linear connection on all tensor fields that satisfies conditions (1)-(5) above. Sketch of proof. First we use the conditions (3)-(5) to derive the formula of r on 1-forms. Let ! 2 Ω1(M) = Γ(T ∗M) be any 1-form, then by (3) and (5) we must have X(!(Y )) = rX (!(Y )) = rX (C(! ⊗ Y )) = C(rX (! ⊗ Y )): Now use (4), we get C(rX (! ⊗ Y )) = C(rX ! ⊗ Y + ! ⊗ rX Y ) = (rX !)(Y ) + !(rX Y ): 1 2 LECTURE 5: THE RIEMANNIAN CONNECTION So we conclude (1.1) (rX !)(Y ) = X(!(Y )) − !(rX Y ); Second we can use (4) iteratively to show that for any (r; s)-tensor field T , (rX T )(!1; ··· ;!r;Y1; ··· ;Ys) =X(T (!1; ··· ;!r;Y1; ··· ;Ys)) X − T (!1; ··· ; rX !i; ··· ;!r;Y1; ··· ;Ys) (1.2) i X − T (!1; ··· ;!r;Y1; ··· ; rX Yj; ··· ;Ys): j This can be done by induction. In particular, this shows the uniqueness. To prove the existence, one just need to check that the connections defined by equations (1.1) and (1.2) satisfies all conditions (1)-(5). In particular, since a Riemannian metric g is a (0; 2)-tensor field on M, we get (rX g)(Y; Z) = XhY; Zi − hrX Y; Zi − hY; rX Zi: Definition 1.2. A tensor field T is called parallel if rX T = 0 for all X 2 Γ(TM). Example. One can view the identity map I = Id : Γ(TM) ! Γ(TM) as a (1; 1)-tensor via I(!; Y ) = !(Y ): Then it is parallel since according to (1.1), (rX I)(!; Y ) = X(!(Y )) − (rX !)(Y ) − !(rX Y ) = 0: 2. The Levi-Civita connection Now let (M; g) be a Riemannian manifold, and r a linear connection on M. Definition 2.1. We say r is compatible with g if the Riemannian metric g is parallel. In other words, r is compatible with g if for all X; Y; Z 2 Γ(TM), X(hY; Zi) = hrX Y; Zi + hY; rX Zi: Definition 2.2. A connection r is on (M; g) is called a Levi-Civita connection (also called a Riemannian connection) if it is torsion-free and is compatible with g. n For example, if we let M = R with the canonical Riemannian metric g0, then l the canonical linear connection (i.e. the one with all Christoffel symbols Γij = 0) is a Levi-Civita connection. An nontrivial example is LECTURE 5: THE RIEMANNIAN CONNECTION 3 n Example. Let M = S equipped with the round metric g = ground, i.e. the induced metric from the canonical metric in Rn+1. We denote by r the canonical (Levi- Civita) connection in Rn+1. For any X; Y 2 Γ(T ∗Sn), one can extend X; Y to smooth vector fields X¯ and Y¯ on Rn+1, at least near Sn. By localities we proved last time, the vector ¯ rX¯ Y at any point p 2 Sn depends only on the vector X¯(p) = X(p) and the vectors X¯(q) = X(q) for q 2 Sn. In other words, it is indepedent of the choice of the ¯ extension we chose. So for simplicy we will write rX Y instead of rX¯ Y for points n n on S . It is a vector that is not necessary tangent to S . We define rX Y be the n \orthogonal projection" of rX Y onto the tangent space of S , i.e. rX Y := rX Y − hrX Y; ~ni~n; where ~n (=~x) is the unit out normal vector on Sn. I claim that it is a (=the) Levi-Civita connection of (M; g). To prove this, first notice that r is bilinear, and rfX Y = frX Y . Also rX (fY ) = rX (fY ) − hrX (fY ); ~ni~n = frX (Y ) − fhrX (Y ); ~ni~n + (Xf)Y − h(Xf)Y; ~ni~n = (Xf)Y + frX Y; where we used the fact that Y is a tangent vector field of Sn and thus h(Xf)Y; ni = 0. So r is a linear connection on Sn. This connection is torsion free because (we use [X; Y ] ? ~n here!) rX Y − rY X = rX Y − rY X − hrX Y − rY X; ~ni~n = [X; Y ] − h[X; Y ]; ~ni~n = [X; Y ]: Finally this connection is compatible with the metric g, since XhY; Zi = hrX Y; Zi + hY; rX Zi = hrX Y; Zi + hY; rX Zi; where we used the fact that the difference between rX Y and rX Y is a vector in the normal direction, and thus is perpendicular to Z. Remark. By the same argument, one can prove that if (X; g) is a Riemannian mani- fold, with a Levi-Civita connection rM , and if (N; ι∗g) is a Riemannian submanifold of (M; g), then the \orthogonal projection" of rM onto TN, N M ¯ T rX Y := (rX¯ Y ) ; defines a Levi-Civita connection on (N; ι∗g). Remark. Since any Riemannian manifold can be embedded to the standard Euclidian space isometrically, the arguments in the previous remark immediately implies that on any Riemannian manifold, there exists Levi-Civita connection! 4 LECTURE 5: THE RIEMANNIAN CONNECTION Our main theorem is to prove Theorem 2.3 (The fundamental theorem of Riemannian geometry). On any Rie- mannian manifold (M; g), there is a unique Levi-Civita connection. Remark. Roughly speaking, smooth manifolds are the underlying space, and in ge- ometry we are interested in various extra geometric structures defined on manifold. Given any smooth manifold, one has • infinitely many different distance function (all compatible with the underly- ing topology), • infinitely many different measures, • infinitely many differnt Riemannian structures, • infinitely many linear connections etc. However, for the first five lectures in this course, we proved that if you fix a Rie- mannian metric, then you will get • a canonical distance function (the Riemannian distance) • a canonical measures (the Riemannian measure), • a canonical linear connections (the Levi-Civita connection). First proof (coordinate free). Assume the Levi-Civita connection exists. Then hrX Y; Zi =X(hY; Zi) − hY; rX Zi =X(hY; Zi) − hY; rZ Xi − hY; [X; Z]i =X(hY; Zi) − Z(hY; Xi) + hrZ Y; Xi − hY; [X; Z]i =X(hY; Zi) − Z(hY; Xi) + hrY Z; Xi + h[Z; Y ];Xi − hY; [X; Z]i =X(hY; Zi) − Z(hY; Xi) + Y (hZ; Xi) − hZ; rY Xi + h[Z; Y ];Xi − hY; [X; Z]i =X(hY; Zi) − Z(hY; Xi) + Y (hZ; Xi) − hZ; rX Y i − hZ; [Y; X]i + h[Z; Y ];Xi − hY; [X; Z]i It follows that rX Y must be the vector satisfying 2hrX Y; Zi =X(hY; Zi) − Z(hY; Xi) + Y (hZ; Xi) (2.1) − hZ; [Y; X]i + h[Z; Y ];Xi − hY; [X; Z]i: The right hand side is determined by the metric. So the uniqueness is proved. [The last formula is called the Koszul formula.] To prove the existence, one only need to check that the rX Y defined by the above formula satisfies all conditions of Levi-Civita connections. Second proof (local coordinate). Again we first prove uniqueness. Let r be a Levi- k Civita connection. Pick a coordinate neighborhood and let Γij be the functions LECTURE 5: THE RIEMANNIAN CONNECTION 5 k k r@i @j = Γij@k. Then it is enough to prove that the Γij's are determined by the metric g. First we note that by torsion free property, k k Γij = Γji: Second we calculate @igjk = @i(g(@j;@k)) = g(r@i @j;@k) + g(@j; r@i @k) l l l l = g(Γij@l;@k) + g(@j; Γik@l) = Γijglk + Γikgjl: Similarly one can prove l l l l @jgki =Γjkgli + Γjigkl and @kgij =Γkiglj + Γkjgil: So we get l @jgki + @igjk − @kgij = 2glkΓij: It follows l lk (2.2) 2Γij = g (@jgki + @igjk − @kgij): This proves the uniqueness. i j For the existence, we can define locally (for X = X @i and Y = Y @j) i j i j l rX Y = X @iY @j + X Y Γij@l; l where Γij is the function given by (2.2).
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