DIFFERENTIAL GEOMETRY, LECTURE 15, JULY 10

Transcription

1 DIFFERENTIAL GEOMETRY, LECTURE 15, JULY Levi-Civita connection From now on we are intereste in connections on the tangent bunle T X of a Riemanninam manifol (X, g). Out main result will be a construction of a canonical connection on T X epening of the Riemannian tensor g Compatibility with the metric. Recall that any connection on T X efines operators of parallel transport Ψ γ : T γ(a) X T γ(b) X Definition. A connection on T X is calle to be compatible with g if the parallel transport operators preserve the metric: v, w = Ψ(v), Ψ(w). Let us fin out when oes this happen in terms of. Recall that a vector fiel s along γ is calle parallel if s = 0. If is compatible t with the metric, each pair of parallel vector fiels s, t along γ has a constant inner prouct. Recall that for any v T x (X) there exists a unique parallel vector fiel s along γ satisfying s(x) = v Lemma. The connection on T X is compatible with the metric iff for any vector fiels v, w along γ one has (1) w v, w = v, w + v, t t t. Proof. Assume is compatible with the metric. Let γ be a curve connecting x with y. Choose an orthogonal basis e 1,..., e n in T x X. Denote the the same letters the parallel vector fiels along γ efine by the e i. Since is compatible with the metric, the vector fiels e i along γ are orthogonal at all points. Now if v = a i e i an w = b i e i where a i, b i are smooth functions on γ, we have v, w = a i b i, v = a i e t t i an w = b i e t t i. The comparison yiels one of the implications. The converse implication is obvious Theorem. A connection is compatible with the metric iff for all τ, v, w T one has (2) τ( v, w ) = τ (v), w + v, τ (w). 1

2 2 Proof. Let be compatible. Then, on orer to check the equality of the functions at a given point x, we have to choose any curve passing through x with tangent vector at x equal to τ x ; restrict vector fiels v an w to γ an apply Lemma To prove the opposite claim we have to euce compatibility of a connection satisfying the property (2). Or, equivalently, this means that the connection satisfies (1) for any curve γ for all pairs of vector fiels on γ inuce by global vector fiels on X. Now we note that it is enough to check the claim for curves γ lying completely insie an open set U such that T X is trivial on U. For such small curves any vector fiel along γ can be presente as a linear combination α i v i where v i age global vector fiels an α i are functions on γ. Now, if v = α i v i an w = β j w j so that v i an w j are global vector fiels, then t v, w = t ( i,j α i β j v i, w j ) = i,j t (α iβ j ) v i, w j + i,j α i β j t v i, w j whereas the right-han sie is i,j α i t β j v i, w j + i,j which is obviously the same. α i β j v i t, w j + i,j α i β j t v i, w j + i,j v w, w + v, t t = α i β j v i, w j t 5.2. Torsion. Compatibility of a connection with the metric coul be efine for any vector bunle V with a metric g Γ(S 2 V ). The following notion makes sense for connections on T X only Definition. A connection on T X is calle torsion-free if for any pair of vector fiels τ, σ one has σ (τ) τ (σ) = [σ, τ]. Choose a local chart with coorinates x 1,..., x n, so that T X is generate by x i. In what follows we will write i instea of for simplicity. Then Since [ i, j ] = 0, symmetricity implies x i i ( j ) = Γ k i,j k. Γ k i,j = Γ k j,i Lemma. In local coorinates, a connection on T X is torsion-free if an only if the corresponing Christoffel symbols satisfy the coniition Γ k i,j = Γ k j,i. Proof. The only if part has alreay been checke. The if part is a result of an easy calculation.

3 5.3. Levi-Civita connection. Let (X, g) be a Riemannian manifol. Levi- Civita connection is the only connection on T X which is compatible with the metric an torsion-free. This is the claim of the following theorem which is the principal theorem of Differential Geometry Theorem. There is a unique connection on the tangent bunle of a Riemannian manifol (X, g) which is torsion-free an compatible with g. Proof. It suffices to prove the existence an uniqueness in local coorinates: because of the uniqueness the connections on ifferent charts will coincie at the intersections. We will enote as usual g i,j = i, j, By compatibility with the metric i ( j ) = k Γ k i,j k. (3) i g j,k = i j, k + j, i k. Replacing the triple (i, j, k) with (j, i, k) an (k, i, j) we get two more equations, (4) j g i,k = j i, k + i, j k. an (5) k g i,j = k i, j + i, k j. Since is to be torsion-free, threre are only three ifferent expressions on the right-han sie of the equations. This allows to express each one of them through the left-han sie as follows. (6) i ( j ), k = 1 2 ( ig j,k + j g i,k k g i,j ). The formula (6) uniquely efines i ( j ) since the inner prouct is nonegenerate. This proves uniqueness of the connection satisfying the liste above properties. To prove the existence of such a connection, we can efine a connection by the formulas (6) an then to check the properties. Torsion-freeness follows irectly from the efinition. The formulas (3) (5) can be immeiately euce from (6). This implies compatibility of with g in the generators. We have alreay seen in the proof of Theorem that this implies the compatibility in general Comparison It is instructive to compare Levi-Civita connections on an embee pair of Riemannian manifols. Let Y be a submanifol of a Riemannian manifol X. The tangent space T y Y is embee into T y X an, therefore, it carries an inuce inner prouct. This gives a Riemannian structure on Y.

4 4 Let i : Y X be the embeing. One has a map of vector bunles j : T Y i (T X) an the connection on i (T X) efine by the Levi-Civita connection on T X. Orthogonal projections π y : T y X T y Y efine a map π : i (T X) T Y splitting j. Now we are reay to express the Levi-Civita connection on T Y through. Theorem. Let σ, τ be vector fiels on Y. Then (7) τ (σ) = π( j(τ) (j(σ))). Proof. The right-han sie of the formula (7) efines a connection on T Y. Let us check it is torsion-free an compatible with the metric. Compatibility with the metric follows from the fact that j is an isometry. Torsion freeness can be checke locally; one can use local coorinates x 1,..., x n for which Y is given by the equations x 1 =... = x i = 0. In this case torsion freeness is immeiate Connection on the stanar tensor bunles. There is a general way to exten a connection on V to a connection on any tensor power T p (V) T q (V ). It is one very similarly to the efinition of Lie erivative on the stanar bunles. More precisely, we claim that, given a connection on V, there is a unique collection of connections (enote by the same letter ) on V p q := T p (V) T q (V ) so that τ acts as τ on V 0 0 = 1. τ (s t) = τ (s) t + s τ (t). τ commutes with the contractions. Instea of giving a proof (which is ientical to the proof given for Lie ervativee) we will present some formulas - for V p 0 an V 0 q. (8) τ (s 1 s p ) = i s 1... τ (s i )... s p, s 1... s p Γ(V p 0 ). (9) ( τ r)(s 1,..., s q ) = τ(r(s 1,..., s q )) i r(s 1,..., τ (s i ),..., s q ). s Γ(V 0 q). Of course, all sai above can be applie to V = T X an the Levi-Civita connection Moving inices up an own Linear algebra Let (V, g) be a vector space enowe with an inner prouct g. Any quaratic form on V can be interprete as a linear map V V.

5 Since g is non-egenerate, the map V V is an isomorphism. It is given by the formula v g(v, ) Similarly to the above, the Riemannian metric g T2 0 inuces an isomorphism T X T X. We will write this map as an isomorphism T0 1 T1 0 (converting vectors to covectors). This can be automatically extene to maps Tq p T p r q+r (moving own r inices. All these maps are isomorphisms, so that one can really move maps up an own. Homework. 0. Check that the formula τ (f) = τ(f) efines a connection on the trivial bunle Let 0 an 1 be two connections on a vector bunle V. Prove that the ifference 1 0 is a tensor, that is is iven by a map T X V V of vector bunles. 2. Let be a connection on T X. Define T : T T T by the formula T (σ, τ) = σ (τ) τ (σ) [σ, τ]. Prove that T is a (1, 2)-tensor, that is is efine by a map T X T X T X of vector bunles. (The tensor T is calle the torsion of ). 3. Prove that if is the Levi-Civita connection on (X, g) then for any τ T one has τ (g) = 0. Here g is consiere as an element of T2. 0 Deuce from this fact that Levi-Civita connection commutes with moving inices up an own.