《黎曼几何》课程教学资源（英文讲义）Lec06 THE LEVI-CIVITA CONNECTION.pdf_大学文库

4 LECTURE 6: THE LEVI-CIVITA CONNECTION ¶ Compatibility of the induced linear connection. Now let M be a smooth manifold, and ∇ a linear connection on (the tangent bundle of) M. As we have seen, ∇ induces linear connections on all tensor bundles ⊗r,sTM over M. It turns out that the induced connections are consistent in the sense that they are compatible with the two natural operations on tensors: the tensor product and the contraction. To see this, let’s consider two examples: Example. For any Y ∈ Γ ∞(TM) and ω ∈ Ω 1 (M) = Γ∞(T ∗M), applying ∇ to the (1, 1)-tensor field Y ⊗ ω we get ∇X(Y ⊗ω)(η, Z) = X(η(Y )ω(Z))−(∇Xη)(Y )ω(Z)−η(Y )ω(∇XZ) = X(η(Y ))ω(Z)−(∇Xη)(Y )ω(Z)+η(Y )X(ω(Z))−η(Y )ω(∇XZ) = ((∇XY )⊗ω)(η, Z)+(Y ⊗(∇Xω))(η, Z). In other words, ∇X(Y ⊗ω) = (∇XY )⊗ω+Y ⊗(∇Xω). Example. Here is another way to understand the fact ∇I = 0: Let C 1 1 be the contraction map that pairs the first tangent component to the first cotangent component, then X(ω(Y )) = ∇X(C 1 1 (Y ⊗ ω)), and by the previous example, C 1 1 (∇X(Y ⊗ ω)) = C 1 1 ((∇XY ) ⊗ ω + Y ⊗ ∇Xω) = ω(∇XY ) + (∇Xω)(Y ). In other words, the fact “the identity map I being parallel” implies the fact “∇ commutes with C 1 1 ” for (1, 1)-tensor. Similarly one can show that for an (r, s)- tensor, ∇ commutes with all contraction C i j ’s. Now we can state the compatibility of ∇ with the two tensor operations: Theorem 1.3. Given a linear connection ∇ on TM, the induced linear connection ∇ : Γ∞(TM) × Γ ∞(⊗ r,sTM) → Γ ∞(⊗ r,sTM), (X, T) 7→ ∇XT, on tensor bundles ⊗r,sTM above is compatible with the tensor product operation 1 (6) ∇X(T1 ⊗ T2) = (∇XT1) ⊗ T2 + T1 ⊗ (∇XT2) and commutes with the contractions (7) C i j (∇XT) = ∇XC i j (T), where 1 ≤ i ≤ r, 1 ≤ j ≤ s, and C i j : Γ∞(⊗ r,sTM) → Γ ∞(⊗ r−1,s−1TM) is the contraction map that pairs the i-th vector with the j-th covector. 1This fact has another beautiful explanation: For any X, the covariant derivative operator ∇X is a derivation on the (graded tensor) algebra of all tensor fields on M!

LECTURE 6: THE LEVI-CIVITA CONNECTION 5 The proof is merely a simple but messy computation which we will omit. Instead, we will show how do we recover (3) using compatibility conditions (6) and (7): ∇X(ω(Y ))=∇X(C 1 1 (Y ⊗ω))=C 1 1 (∇X(Y ⊗ω)) =C 1 1 (Y ⊗∇Xω + ∇XY ⊗ω)= (∇Xω)(Y ) + ω(∇XY ) which is another way to write (3). Moreover, by a tedious messy induction argument in the same philosophy, one can even recover (5) by using (6) and (7). In other words,one has Theorem 1.4. Given any linear connection ∇ on the tangent bundle TM, there is a unique linear connection on all tensor fields that coincides with ∇ on TM, coincides with (1) on functions, and satisfies compatibility conditions (6) and (7) above. ¶ The Hessian of a function. Let M be a smooth manifold and ∇ a linear connection on M. One may equivalently write the induced linear connections on tensor bundles as maps ∇ : Γ∞(⊗ r,sTM) → Γ ∞ T ∗M ⊗ (⊗ r,sTM) = Γ∞(⊗ r,s+1TM) with the understanding that ∇T(· · · , X) = (∇XT)(· · ·). Then one may iterate ∇ to get ∇2 : Γ∞(⊗ r,sTM) → Γ ∞(⊗ r,s+2TM) (or even higher order powers) in the understanding that (8) ∇2T(· · · , X, Y ) = (∇Y ∇T)(· · · , X) = (∇Y ∇XT)(· · ·) − (∇∇Y XT)(· · ·). [Note that ∇2T(· · · , X, Y ) ̸= (∇Y ∇XT)(· · ·) in general.] In particular, if we take r = s = 0, i.e. consider functions f ∈ C ∞(M), we get ∇2 f(X, Y ) = (∇Y df)(X) = Y Xf − (∇Y X)f. The bilinear form ∇2 f is known as the Hessian of f with respect to ∇. ¶ Torsion tensors of a linear connection. For a general linear connection ∇, the Hessian is not interesting, since it might be non-symmetric. A natural question is: when will ∇2 f symmetric? We calculate: ∇2 f(X, Y ) − ∇2 f(Y, X) = (∇XY )f − (∇Y X)f − XY f + Y Xf = (∇XY − ∇Y X − [X, Y ])f It follows that the vector field (9) T (X, Y ) = ∇XY − ∇Y X − [X, Y ] measures how far ∇2 f from being symmetric. A direct computation shows T (fX, Y ) = T (X, fY ) = fT (X, Y )

LECTURE6:THELEVI-CIVITACONNECTION72.THELEVI-CIVITA CONNECTIONWe know that for any smooth manifolds, there are numerous choices of metricstructures and measure structures. But with a Riemannian metric structure g athand,one can define a unique canonical metric structure and measure structureassociated to g. The same phenomena happens for linear connections: Given aRiemannian metric g, there a unique canonical linear connection associated to g,known as the Levi-Civita connection,which hasmany niceproperties.I Metric compatible linear connection.Let (M,g) be a Riemannian manifold. Before we write down the definition ofthe Levi-Civita connection, we may ask ourselves a question:what kind of niceproperties do we want?First, we may want the linear connection to be torsion free, since under thiscondition, the Hessian is symmetric (and in fact as we will see later, there will bemany other nice symmetry properties under the torsion free condition). Second, wemay want the linear connection to be“compatible with the Riemannian metric g"Now a natural question is:Question:when we say a linear connection is"compatible with theRiemannian metric g", what do we really mean?Let's explore this question from the geometric point of view. With a Riemannianmetric g at hand, we get an inner product on each T,M. So a natural requirementwould beAnswer: we may require that each parallel transportPot : (T(0); 9%(0)) → (T(t); 9(t)preserves the given inner product structure (i.e. is an isometry betweenthe two inner product spaces).It turns out that the geometric requirement above is equivalent to an algebraicequation on Vx (which is easy to use) and also to an analytic equation on g:Proposition 2.1. Let be a linear connection on a Riemannian manifold (M,g).Then the following statements are equivalent:(1) All the parallel transports Po.t : (T(0), (o) → (Tr(t),r(t)) are isometries.(2) For any smooth vector fields X,Y,Z E To(TM), one hasX((Y,Z)) = (VxY,Z) + (Y, VxZ)(3) g is parallel, i.e. Vg =0.Proof. (1) -→ (2)Let be a linear connection such that Po, are isometries. Forany vector fields X,Y,z I(TM), and any p M, take a curve such that(O) = p and (O) = Xp. Take an orthonormal basis [ei) of (T,M, gp), and let ei;(t)

LECTURE 6: THE LEVI-CIVITA CONNECTION 7 2. The Levi-Civita connection We know that for any smooth manifolds, there are numerous choices of metric structures and measure structures. But with a Riemannian metric structure g at hand, one can define a unique canonical metric structure and measure structure associated to g. The same phenomena happens for linear connections: Given a Riemannian metric g, there a unique canonical linear connection associated to g, known as the Levi-Civita connection, which has many nice properties. ¶ Metric compatible linear connection. Let (M, g) be a Riemannian manifold. Before we write down the definition of the Levi-Civita connection, we may ask ourselves a question: what kind of nice properties do we want? First, we may want the linear connection to be torsion free, since under this condition, the Hessian is symmetric (and in fact as we will see later, there will be many other nice symmetry properties under the torsion free condition). Second, we may want the linear connection to be “compatible with the Riemannian metric g”. Now a natural question is: Question: when we say a linear connection is “compatible with the Riemannian metric g”, what do we really mean? Let’s explore this question from the geometric point of view. With a Riemannian metric g at hand, we get an inner product on each TpM. So a natural requirement would be Answer: we may require that each parallel transport P γ 0,t : (Tγ(0), gγ(0)) → (Tγ(t) , gγ(t)) preserves the given inner product structure (i.e. is an isometry between the two inner product spaces). It turns out that the geometric requirement above is equivalent to an algebraic equation on ∇X (which is easy to use) and also to an analytic equation on g: Proposition 2.1. Let ∇ be a linear connection on a Riemannian manifold (M, g). Then the following statements are equivalent: (1) All the parallel transports P γ 0,t : (Tγ(0), gγ(0)) → (Tγ(t) , gγ(t)) are isometries. (2) For any smooth vector fields X, Y, Z ∈ Γ ∞(TM), one has X(⟨Y, Z⟩) = ⟨∇XY, Z⟩ + ⟨Y, ∇XZ⟩. (3) g is parallel, i.e. ∇g = 0. Proof. (1) =⇒ (2) Let ∇ be a linear connection such that P γ 0,t are isometries. For any vector fields X, Y, Z ∈ Γ ∞(TM), and any p ∈ M, take a curve γ such that γ(0) = p and ˙γ(0) = Xp. Take an orthonormal basis {ei} of (TpM, gp), and let ei(t)

8 LECTURE 6: THE LEVI-CIVITA CONNECTION be the parallel transport of ei along γ. By assumption, {ei(t)} is an orthonormal basis at γ(t). If we denote Y = Y i (t)ei(t) and Z = Z i (t)ei(t), then ⟨Y, Z⟩ = XY i (t)Z i (t) along γ. So ∇Xp ⟨Y, Z⟩ = XXp(Y i (t))Z i (0) + Y i (0)Xp(Z i (t)) = ⟨∇Xp Y, Zp⟩ + ⟨Yp, ∇XpZ⟩, i.e. ∇ satisfies the desired equation. (2) =⇒ (1) Conversely, suppose ∇ be a linear connection on M such that X(⟨Y, Z⟩) equals ⟨∇XY, Z⟩+⟨Y, ∇XZ⟩ for all X, Y, Z. Fix any curve γ, let {ei} be an orthonormal basis at p = γ(0), and let ei(t) be the parallel transport of ei along γ, then d dt⟨ei(t), ej (t)⟩ = ˙γ(t)(⟨ei(t), ej (t)⟩) = ⟨∇γ˙ (t)ei(t), ej (t)⟩ + ⟨ei(t), ∇γ˙ (t)ej (t)⟩ = 0. It follows that {ei(t)} remains to be an orthonormal basis for (Tγ(t) , gγ(t)). So the linear map P γ 0,t is an isometry. (2) ⇐⇒ (3) Recall that the Riemannian metric g is a (0, 2)-tensor, and thus one can take its covariant derivative ∇g, which by definition is given by (∇Xg)(Y, Z) = X⟨Y, Z⟩ − ⟨∇XY, Z⟩ − ⟨Y, ∇XZ⟩. So the conclusion follows. □ Definition 2.2. We say a linear connection ∇ on a Riemannian manifold (M, g) is compatible with g if one (and thus all) of the three equivalent conditions in Proposition 2.1 hold. Remark. Note that by the geometric condition, if ∇ is a metric-compatible linear connection on (M, g), and if X, Y are vector fields parallel along a curve γ, then ⟨X, Y ⟩ is a constant on γ. ¶ The Levi-Civita connection. Finally we define Definition 2.3. A connection ∇ 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. We example two simple examples: Example. Let M = R m, equipped with the canonical Riemannian metric g0, then the canonical linear connection (i.e. the one with all Christoffel symbols Γl ij = 0 under the canonical basis) is a Levi-Civita connection. Example. Equip M = S m with the round metric g = ground, i.e. the induced metric from the canonical metric in R m+1. We denote by ∇ the canonical (Levi-Civita)

LECTURE 6: THE LEVI-CIVITA CONNECTION 9 connection in R m+1. For any X, Y ∈ Γ ∞(T Sm), one can extend X, Y to smooth vector fields X¯ and Y¯ on R m+1. By localities we proved last time, the vector ∇X¯Y¯ at any point p ∈ S m depends only on the vector X¯(p) = X(p) and the vectors X¯(q) = X(q) for q ∈ S m near p. In other words, it is independent of the choice of the extension. So for simplicity we will write ∇XY instead of ∇X¯Y¯ for points on S m. It is a vector that is not necessary tangent to S m. We define ∇XY be the “orthogonal projection” of ∇XY onto the tangent space of S m, i.e. ∇XY := ∇XY − ⟨∇XY, ⃗n⟩⃗n, where ⃗n = (x 1 , x2 , · · · , xm+1) is the unit out normal vector on S m. Observe that ∇X⃗n = X i ∂i(x j )∂j = X. It follows ⟨∇XY, ⃗n⟩⃗n = −⟨Y, ∇X⃗n⟩⃗n = −⟨X, Y ⟩⃗n and thus ∇XY = ∇XY + ⟨X, Y ⟩⃗n. We claim that ∇ is a Levi-Civita connection of (S m, ground). To prove this, first notice that ∇ is bilinear, and ∇fXY = f∇XY . Also ∇X(fY ) = ∇X(fY ) + ⟨X, fY ⟩⃗n = (Xf)Y + f∇XY + f⟨X, Y ⟩⃗n = (Xf)Y + f∇XY. This connection is torsion free because ∇XY − ∇Y X = ∇XY + ⟨X, Y ⟩⃗n − ∇Y X − ⟨Y, X⟩⃗n = ∇XY − ∇Y X = [X, Y ]. Finally this connection is compatible with the metric g, since X⟨Y, Z⟩ = ⟨∇XY, Z⟩ + ⟨Y, ∇XZ⟩ = ⟨∇XY, Z⟩ + ⟨Y, ∇XZ⟩, where we used the fact that Z is perpendicular to ⃗n. Remark. If (M, g) is a Riemannian manifold, with a Levi-Civita connection ∇M, and if (N, ι∗ g) is a Riemannian submanifold of (M, g), then we can define a connection on N by the same trick, namely orthogonally project ∇M onto T N, ∇N XY := (∇M X¯ Y¯ ) T , One can prove that it is the Levi-Civita connection on (N, ι∗ g). ¶ The fundamental theorem of Riemannian geometry. Since any Riemannian manifold can be embedded to the standard Euclidian space isometrically, the arguments in the previous remark implies that on any Riemannian manifold, there exists a Levi-Civita connection! In what follows we will give two direct elementary proofs of this fact, and also prove the uniqueness: Theorem 2.4 (The fundamental theorem of Riemannian geometry). On any Riemannian manifold (M, g), there is a unique Levi-Civita connection

10 LECTURE 6: THE LEVI-CIVITA CONNECTION First proof (local coordinate). We first prove uniqueness. Let ∇ be a Levi-Civita connection. Pick a coordinate chart and let Γk ij be the Christoffel symbols. It is enough to prove that the Γk ij ’s are determined by gij ’s. The trick already appeared in Lecture 1. First we note that by torsion free property, Γk ij = Γk ji. Second we calculate ∂igjk = ∂i(g(∂j , ∂k)) = g(∇∂i∂j , ∂k) + g(∂j , ∇∂i∂k) = g(Γl ij∂l , ∂k) + g(∂j , Γ l ik∂l) = Γl ijglk + Γl ikgjl. Similarly one can prove ∂jgki =Γl jkgli + Γl jigkl and ∂kgij =Γl kiglj + Γl kjgil. So we get ∂jgki + ∂igjk − ∂kgij = 2glkΓ l ij . It follows (10) 2Γl ij = g lk(∂jgki + ∂igjk − ∂kgij ). This proves the uniqueness. [This is essentially the same as we did in Lecture 1.] For the existence, we can define locally (for X = Xi∂i and Y = Y j∂j ) ∇XY = X i (∂iY j )∂j + X iY jΓ l ij∂l , where Γl ij is the function given by (10). By tedious computations one can check that this give a Levi-Civita connection whose Christoffel symbols are the Γl ij ’s. □ Second proof (coordinate free). Again we first prove the uniqueness. Assume the Levi-Civita connection exists. Then (use torsion-free and metric-compatibility in turns) ⟨∇XY, Z⟩ =X(⟨Y, Z⟩) − ⟨Y, ∇XZ⟩ =X(⟨Y, Z⟩) − ⟨Y, ∇ZX⟩ − ⟨Y, [X, Z]⟩ =X(⟨Y, Z⟩) − Z(⟨Y, X⟩) + ⟨∇ZY, X⟩ − ⟨Y, [X, Z]⟩ =X(⟨Y, Z⟩) − Z(⟨Y, X⟩) + ⟨∇Y Z, X⟩ + ⟨[Z, Y ], X⟩ − ⟨Y, [X, Z]⟩ =X(⟨Y, Z⟩) − Z(⟨Y, X⟩) + Y (⟨Z, X⟩) − ⟨Z, ∇Y X⟩ + ⟨[Z, Y ], X⟩ − ⟨Y, [X, Z]⟩ =X(⟨Y, Z⟩) − Z(⟨Y, X⟩) + Y (⟨Z, X⟩) − ⟨Z, ∇XY ⟩ − ⟨Z, [Y, X]⟩ + ⟨[Z, Y ], X⟩ − ⟨Y, [X, Z]⟩. It follows that ∇XY must be the vector satisfying (11) 2⟨∇XY, Z⟩ =X(⟨Y, Z⟩) − Z(⟨Y, X⟩) + Y (⟨Z, X⟩) − ⟨Z, [Y, X]⟩ + ⟨[Z, Y ], X⟩ − ⟨Y, [X, Z]⟩. The right hand side is determined by the metric. So the uniqueness is proved. [(11) is called the Koszul formula, which reduce to (10) if we take X, Y, Z to be ∂i , ∂j and ∂l .] To prove the existence, one “only need” to check that the ∇XY defined by the above formula satisfies all conditions of Levi-Civita connections. □