The Euclidean space
We claim that the Levi-Civit`a connection∇inRn coincides with the usual derivative. In fact, let (x1, . . . , xn) denote the standard global coordinates inRn. We have that
g ∂
∂xi, ∂
∂xj
=δij and ∂
∂xi, ∂
∂xj
= 0
for all i,j. Plugging these relations into the Koszul formula (2.2.6) gives that ∇ ∂
∂xi
∂
∂xj = 0 for all i,j, namely, all the Christoffel symbols Γijk = 0. If
X =X
j
aj ∂
∂xj and Y =X
k
bk ∂
∂xk,
forai,bj ∈C∞(Rn), then, using formula (2.2.2),
∇XY =X
i
X
j
aj∂bi
∂xj
∂
∂xi =X(Y) =dY(X),
proving the claim. We also get, from equation (2.3.4), that a vector field X along a curve γ : [a, b]→M, given as
X(t) =X
k
ak(t) ∂
∂xk γ(t),
is parallel if and only theak are constant functions, namely, the parallel vector fields inRnare the constant vector fields. It follows that the parallel transport map along γ from a to b is given by the differential of the translation map, that is,
Pb,aγ =d(τv)γ(a),
whereτv is the translation in Rn by the vectorv=γ(b)−γ(a), and, in particular, is independent of the curve γ joiningγ(a) andγ(b). Finally, the geodesic equation (2.4.1) inRn is
(xi)′′= 0 for all i, so the geodesics are the lines. Hence
expp(v) =p+v forp∈Rn and v∈TpRn=Rn.
Product Riemannian manifolds
Let (Mi, gi), wherei= 1, 2, denote two Riemannian manifols and consider the product Riemannian manifold (M, g) = (M1, g1)×(M2, g2). LetUi ∈Γ(T Mi), wherei= 1, 2, be arbitrary vector fields.
Of course,U1andU2 can be identified with vector fields onM, and it follows from the construction of (M, g) that [U1, U2] = 0 and g(U1, U2) = 0 in M.
Now, suppose that X,Y,Z ∈Γ(T M) can be decomposed as X=X1+X2 and Y =Y1+Y2, Z =Z1+Z2, where Xi, Yi, Zi ∈Γ(T Mi) fori= 1, 2 (not every vector field on M admits such a decomposition!). Note that
Xg(Y, Z) =X1g1(Y1, Z1) +X2g2(Y2, Z2) and
g([X, Y], Z) =g1([X1, Y1], Z1) +g2([X2, Y2], Z2).
It then follows from the Koszul formula (2.2.6) applied three times that g(∇XY, Z) = g1(∇1X1Y1, Z1) +g2(∇2X2Y2, Z2)
= g(∇1X1Y1+∇2X2Y2, Z),
where∇denotes the Levi-Civit`a connection ofM and∇idenotes the Levi-Civit`a connection ofMi fori= 1, 2. Sincegis nondegenerate and any tangent vector toM can be extended to a vector field
Z which decomposes asZ1+Z2, this calculation yields the following formula for the Levi-Civit`a connection of a Riemannian product:
(2.8.1) ∇XY =∇1X1Y1+∇2X2Y2.
It follows from this formula that the Christoffel symbol Γijk of ∇ is zero unless all the three indicesi,j,kcorrespond to coordinates of the same factorMℓ, where ℓ= 1 or 2, in which case Γijk is a function onMℓand a Christofell symbol of∇ℓ. Therefore ifγ is a curve inM with components γ1 in M1 and γ2 in M2, and X is a vector field along γ, then we can decompose X = X1+X2 whereXi is a vector field alongγi, and equation (2.3.2) gives ∇dtX = ∇dtX1 +∇dtX2. In particular, X is parallel along γ if and only if Xi is parallel along Mi for i = 1, 2. Asγ′(t) = γ1′(t) +γ2′(t), in particular yet,γ is a geodesic if and only ifγi is a geodesic of Mi fori= 1, 2.
Riemannian submanifolds and isometric immersions
Let (M, g), (M , g) be Riemannian manifolds, and suppose thatι:M →M is an isometric immer-sion. We would like to relate the Levi-Civit`a connections∇ofM and∇ofM. Since this is a local problem, we can work in a neighborhood a point p ∈M and assume that ι is the inclusion map.
Now the tangent bundleT M is a subbundle of T M, the metricg is the restriction ofg, and every vector field onM admits an extension to a vector field on M.
LetX,Y and Z be vector fields onM, and letX,Y and Z be extensions of those vector fields to vector fields onM. Note that [X, Y] is an extension of [X, Y] to a vector field on M. It follows from two applications of the Koszul formula (2.2.6) that
2g((∇XY)p, Zp) = 2g((∇XY)p, Zp)
= S ±Xpg(Y, Z)±g([X, Y]p, Zp)
= S ±Xpg(Y , Z)±g([X, Y]p, Zp)
= 2g((∇XY)p, Zp)
= 2g((∇XY)p, Zp),
whereSdenotes cyclic summation inX,Y,Z. Since (∇XY)p ∈TpM and Zp can be any element of TpM, it follows that
(2.8.2) (∇XY)p= Πp (∇XY)p
, where Πp:TpM →TpM is the orthogonal projection.
The most important case is that of Riemannian submanifolds of Euclidean space. If M is a Riemannian submanifold of Rn, then formula (2.8.2) implies that a smooth curve γ in M is a geodesic of M if and only if its second derivative γ′′ in Rn is everywhere normal to M; in other words, the geodesics of M are the “curves with normal acceleration”.
The sphere Sn
Let p∈ Sn and v ∈TpSn. We now determine the unique geodesic γ of Sn with initial conditions γ(0) =p and γ′(0) =v. If v = 0, thenγ is a constant curve, so we may assume that v6= 0. Since p and v are orthogonal vectors in Rn+1, they span a 2-dimensional subspace which we denote by E. Let f :Rn+1 → Rn+1 be the linear reflection on E. Then f is an orthogonal transformation of Rn+1 and leaves Sn+1 invariant. Now every orthogonal transformation ofRn+1 is an isometry.
Since Sn+1 has the induced metric from Rn+1, f restricts to an isometry of Sn which we denote
by the same letter. Owing to the fact that an isometry maps geodesics to geodesics, the curve
˜
γ = f ◦γ is a geodesic of Sn. Since f leaves E pointwise fixed, the initial conditions of ˜γ are
˜
γ(0) = f(γ(0)) = f(p) = p and ˜γ′(0) = f(γ′(0)) = f(v) = v, namely, the same as those of γ.
By the uniqueness of geodesics with given initial conditions, we have that ˜γ = γ, or, what is the same,f(γ(t)) =γ(t) for all tin the domain ofγ. It follows that γ is contained inE and thus must coincide with the great circleSn∩E parametrized with constant speed on its domain of definition.
This argument shows that the great circles are locally geodesics; but then, the great circles are geodesics.
In particular, the geodesics ofSn parametrized by arc-length are periodic of period 2π. Finally, we have the formula
expp(v) = cos(||v||)p+ sin(||v||) v
||v||
forv6= 0.
Riemannian coverings
Letπ : ( ˜M ,g)˜ →(M, g) be a Riemannian covering.
2.8.3 Proposition The geodesics of(M, g) are the projections of the geodesics of( ˜M ,˜g), and the geodesics of ( ˜M ,g)˜ are the liftings of the geodesics of (M, g).
Proof. Suppose ˜γ and γ are continuous curves in ˜M, M such that π ◦γ˜ = γ. Since π is a local isometry, it maps a sufficiently small arc of ˜γ isometrically onto a small arc of γ. It follows that ˜γ is a geodesic if and only if γ is a geodesic. This shows that the classes of curves described in the statement of the proposition are indeed geodesics. Now we need only to remark that every continuous curve inM is the projection of any of its continuous liftings in ˜M, and every continuous curve in ˜M is the continuous lifting of its projection toM. The real projective space
We apply Proposition 2.8.3 to the Riemannian covering map π :Sn→RPn. The geodesics of Sn have already been determined as being the great circles parametrized with constant speed, so the geodesics of RPn are the projections of those. In particular, sinceπ identifies antipodal points of Sn, the geodesics of RPn paramerized by arc-length are periodic of periodπ.
Flat tori
Let Γ be a lattice inRnand consider the induced Riemannian metric gΓ on Tn. We apply Propo-sition 2.8.3 to the Riemannian covering map π : Rn → (Tn, gΓ) to deduce that the geodesics of (Tn, gΓ) are simply the projections of the straight lines in Rn. In this way, we see that some geodesics of (Tn, gΓ) are periodic and some are dense inTn.
Next, let Γ′ be another lattice in Rn. We have already remarked that (Tn, gΓ) and (Tn, gΓ′) are generally non-isometric. Nevertheless, there exists a unique affine transformationf ofRn that maps Γ to Γ′, and hence induces a diffeomorphism ¯f :Rn/Γ→Rn/Γ′ such that the diagram
Rn −−−−→f Rn
y
y Rn/Γ −−−−→f¯ Rn/Γ′
is commutative. In general, ¯f is not an isometry, but since f maps straight lines to straight lines, f¯maps the geodesics of (Tn, gΓ) to the geodesics of (Tn, gΓ′). Hence we get an example of two non-isometric metrics on the same smooth manifold with the same geodesics.
Lie groups ⋆
Let G be a Lie group and denote its Lie algebra by g. In this example, we will describe the Levi-Civit`a connection associated to a bi-invariant metric on G. We start with a definition and a proposition.
We say that an inner product h,i ong is ad-invariant if the identity (2.8.4) hadZX, Yi+hX,adZYi= 0
holds for everyX,Y,Z ∈g.
2.8.5 Proposition EveryAd-invariant inner product ongisad-invariant, and the converse holds if G is connected.
Proof. Let h,i be an inner product ong. It being Ad-invariant means that
(2.8.6) hAdgX,AdgYi=hX, Yi
for every g ∈ G and X, Y ∈g. In particular, taking g = exptZ for Z ∈g and differentiating at t= 0 yields identity (2.8.4).
Assume now that Gis connected and h,i is ad-invariant. Then (2.8.4) holds; note that what it is really saying is thatfX,Y′ (0) = 0 for allX,Y ∈g, where
fX,Y(t) =hAdexptZX,AdexptZYi,
and from this information we will show that fX,Y(t) = fX,Y(0). Indeed, since t 7→ AdexptZ is a homomorphism,
fX,Y(t+s) =fX′,Y′(t)
where X′ = AdexpsZX and Y′ = AdexpsZY. Differentiating this identity at t = 0 gives that fX,Y′ (s) =fX′ ′,Y′(0) = 0. Since s∈R is arbitrary, this implies thatfX,Y is constant, as desired.
So far we have shown that (2.8.6) holds if g lies in the image of exp. But there exists an open neighborhoodU of the identity ofGcontained in the image of exp, and it is known thatU generates Gas a group due to the connectedness ofG. Sinceg7→Adgis a homomorphism, this finally implies
that (2.8.6) holds for everyg∈G.
Let g be a bi-invariant metric on G. Now we are ready to use the Koszul formula (2.2.6) to compute the Levi-Civit`a connection on left-invariant vector fields. Let X, Y,Z ∈g. SinceX and Y are left-invariant vector fields andg is a left-invariant metric,g(X, Y) is a constant function on G. Therefore Zg(X, Y) = 0. Similarly, Y g(Z, X) = Zg(X, Y) = 0. Regarding the other terms of (2.2.6), the preceding proposition shows thatg1 is an ad-invariant inner product ong, so (2.8.7) g([Z, X], Y) +g(X,[Z, Y]) =g1(adZX, Y) +g1(X,adZ, Y) = 0.
We deduce that
(2.8.8) ∇XY = 1
2[X, Y]
for all X, Y ∈ g (this formula shows in particular that ∇XY is also a left-invariant vector field, but this fact of course also follows from general properties of isometries, cf. section 2.5). An important application of this formula is that ∇XX = 0 for all ∈ g, and this means that every one-parameter subgroup ofG thorough the identity is a geodesic. This is also equivalent to saying that the exponential map of G qua Lie group and the exponential map of G qua Riemannian manifold (G, g) coincide. It follows from the Hopf-Rinow theorem to be proved in the next chapter that the exponential map of a compact connected Lie group is surjective, see Theorem 3.3.2 and Corollary 3.3.6. Of course, the geodesics of G through an arbitrary point are left-translates of one-parameter subgroups, namely, of the form t7→gexptX forg∈Gand X∈g.