GEODESIC FLOW 1. Introduction in This Lecture Note We Will Introduce the Notion of Geodesics and Geodesic Flow for a Riemannian

Home , Flow (mathematics), Generator (mathematics), Tangent bundle, Tangent space

SHUBHAM SINHA

1. Introduction In this lecture note we will introduce the notion of Geodesics and Geodesic flow for a Riemannian manifold equipped with the Riemann- ian connection. Intuitively a geodesic is a curve with zero acceleration on the manifold M. When the manifold is Rn, then geodesics are precisely the constant velocity curve (c0(t) = v is constant, where tangent space of points in Rn is canonically identified with Rn). For a smooth surface in R3, geodesics are the curves with acceleration c00(t) (in the ambient space R3) perpendicular to the surface. Such characterization was studied by Bernoulli and Euler, but Gauss understood relation between geodesics and curvature of the surface. Geodesics starting at a point p ∈ M are uniquely defined a the starting velocity v ∈ TpM. This statement implies that points in the tangent bundle TM defines geodesics. When we look at the ‘time evolution’ of the geodesics on the tangent bundle TM, we get a local flow (and vector field) on TM which we call Geodesic flow. We will give the rigorous definition in the next section. Geodesic flow enables us to define a canonical exponential map which is diffeomorphism from an open ball in TpM to an open neighborhood of p. I would like to end the introduction by giving an easy example de- scribing geodesic flow. Let M = S1 be a circle defined by x2 + y2 = 1 in R2 with the induced Riemannian metric, then the tangent bundle TM can be identified with the cylinder x2 + y2 = 1 in R3, where v = (p, z) = (x, y, z) ∈ TpM is an element with |v|p = |z| (and counter clockwise is the positive direction). In this setting the Geodesic flow on TM is given by φ : R × TM → TM defined by φ(t, (cos θ0, sin θ0, z)) = (cos(θ0 + tz), sin(θ0 + tz), z). Observe that if π : TM → M is the projection, then π ◦ φ(t, (p, z)) is the geodesic passing through p with velocity z (i.e uniform circular motion with speed |z|). 1 2 SHUBHAM SINHA

2. Preliminary I am rephrasing required results from the book Introduction to smooth manifolds by John.M.Lee. Definition 2.1. Let M be a smooth manifold and X be a smooth vector field on M, then a curve γ : I → M called a trajectory of X if γ0(t) = X(γ(t)) as an element of Tγ(t)M for all t ∈ I. Proposition 2.2. Let X be a smooth vector field on M, then for any point p ∈ M, there exist an and a trajectory γ defined on (−, ) with γ(0) = p. Moreover, γ is unique trajectory starting at p (where we identify γ and γ0 if they agree on a neighborhood of 0). Proof. This a consequence of the existence and uniqueness of the solution of linear system of ordinary differential equations. Definition 2.3. We define a global flow to be a differential map φ : R × M → M which satisfies φ(t + s, p) = φ(t, φ(s, p)). Remark 2.4. A global flow is simply a group action of Lie group R on the manifold M. We can think of global flow as a process of ‘sliding’ points in M. The following proposition lets us define a vector field corresponding to a global flow which is called infinitesimal generator φ. Proposition 2.5. Let φ be a global flow, then X defined by

∂φ(t, p) (1) X(p) = ∂t t=0 is a smooth vector field on M. Moreover for any p ∈ M, the curve φ(t, p) is a trajectory of X at p. The above proposition is not a bijective correspondence. This is because in ODE the existence of the solution to a linear system is local, hence given a vector field we will not be able to obtain a global flow. Definition 2.6. We define flow to be a function φ : D → R × M → M from an open subset D in R×M such that {0}×M ⊂ D, for all p ∈ M we have φ(0, p) = p and satisfy transnational property φ(t + s, p) = φ(t, φ(s, p)) whenever it defined. Proposition 2.7. There is a canonical bijection between the set of flows on M and set of vector field on M, where one way map is given by Equation(1). Remark 2.8. Note that in the proposition above we if the manifold is not compact, there may not be a uniform δ > 0 such that (−δ, δ)×M ⊂ GEODESIC FLOW 3

D. Thus if we want such a property we will have to restrict out domain to some open subsets of M. Theorem 2.9. If X is a smooth vector ﬁeld on the open set V in the manifold M and p ∈ V then there exist an open set V0 ⊂ V , p ∈ V0, a number δ > 0 and a map φ :(−δ, δ) × V0 → V such that the curve t → φ(t, q), t ∈ (−δ, δ), is the unique trajectory of X starting at q, for every q ∈ V0.

3. Geodesic I will be following chapter 3 of’Riemannian geometry’ by Do Carmo for the rest of the notes.

Deﬁnition 3.1. A curve γ : I → M is called a geodesic at t0 ∈ I if D the covariant derivative dt (dγ/dt) = 0 at the point t0. If γ is geodesic at all points t ∈ I, we say γ is a geodesic. Remark 3.2. If γ as above is a geodesic, then by using property of Riemannian connection we have d dγ dγ D dγ dγ , = 2 , = 0. dt dt dt dt dt dt dγ Thus we have that length of dt = c is a constant. Now we are going to determine local equation satisﬁed by geodesic curves. Let (U, x) be a local of local coordinates about γ(t0). In U, a curve γ

γ(t) = (x1(t), . . . xn(t)) is a geodesic iff 2 D dγ X d xk X dxi dxj (2) 0 = = + Γk X dt dt dt2 ij dt dt k k i,j k k where Xk = ∂/∂x in the local coordinates and Γij are the Christoffel symbols. Hence the desired equation satisfied by the curve is a second order system 2 d xk X dxi dxj (3) + Γk = 0. dt2 ij dt dt i,j Note that second order system do not give rise to a description of a vector field. So we use this to define a first order local system on the tangent bundle TM which gives us prescription of a vector field (and flow) on TM. 4 SHUBHAM SINHA

Note that TU = U × Rn over open set U with coordinates system (x1(t), . . . , xn(t), y1(t), . . . yn(t)) and π : TM → M is the projection given by π(q, v) = q. Observe that any curve γ determines a curve t → (γ(t), γ0(t)) in TM. dx1(t) dxn(t) If is a geodesic then in local coordinate t → (x1(t) . . . , xn(t), dt ,... dt ) satisfies the first order system ( dxk = y (4) dt k dyk P k dt = − i,j Γijyiyj on TU. Lemma 3.3. There exist a unique vector field G on TM whose tra- jectories are of the form t → (γ(t), γ0(t)), where γ is a geodesic on M. Proof. The existence is obtained by defining it locally by (4). By com- patibility of Christoffel symbols, we know it is a well defined and it satisfies the required property. From the above discussion, any G with required property must satisfy 4 hence by uniqueness of the first order local system we get the uniqueness of G. Definition 3.4. The vector field defined above is called the geodesic field on TM and its corresponding flow is called the geodesic flow on TM. Applying theorem 2.9 to the geodesic field at point (p, 0) ∈ TM, we obtain Proposition 3.5. For each p ∈ M, there exist an open set U in TU, where (U, x) is a coordinate system at p and (p, 0) ∈ U, a number δ > 0 and a map φ :(−δ, δ) × U → TU, such that t → φ(t, q, v) is the unique trajectory of G which satisfies the initial condition φ(0, q, v) = (q, v) for each (q, v) ∈ TU. Observe that it is possible to choose U in the form

(5) U = {(q, v) ∈ TU; q ∈ U and v ∈ TqM with |v| < 1} , where V ⊂ U is an open subset. Moreover by putting γ = π ◦ φ in the previous proposition we get the following Proposition 3.6. Given p ∈ M, there exist an open set V ∈ M con- taining p, numbers δ > 0 and 1 > 0 and a map (6) γ :(−δ, δ) × U → M GEODESIC FLOW 5

, where U is defined in equation (5) such that the curve t → γ(t, q, v), t ∈ (−δ, δ), is the unique geodesic of M starting at q ∈ V with velocity v ∈ TqM with |v| < 1. Remark 3.7. The above proposition asserts the existence and uniqueness of geodesic γ(t, q, v) for small enough velocity |v| < 1 defined on the time interval (−δ, δ). The next lemma describes that we can increase (decrease) the time interval where a geodesic is defined by re- ducing (increasing) the permissible magnitude of velocity. Lemma 3.8. If the geodesic γ(t, q, v) is defined on the interval (−δ, δ), then the geodesic γ(t, q, av). where a > o is a real number, is defined on the interval (−δ/a, δ/a) and γ(t, q, av) = γ(at, q, v). Proof. Let h :(−δ, δ) → M be given by h(t) = γ(at, q, v), then h(0) = q dh 0 0 and dt (0) = av. Moreover since by chain rule h (t) = aγ (at, q, v), we have D dh 0 2 (7) = ∇ 0 h (t) = a ∇ 0 γ(at, q, v) = 0 dt dt h (t) γ (at,q,v) where the first equality make sense because by extension lemma we can extend h0(t) to a neighborhood of h(t) if h0(t) 6= 0. Otherwise we get v = 0, in which case h is constant. Thus h is a geodesic starting at q with velocity av at t = 0, hence by uniqueness we get h(t) = γ(av, q, v) = γ(v, q, av). Using lemma 3.8, we can uniformly increase the interval of definition in Proposition 3.6 (i.e we can take δ as big as we want at the cost of maximum velocity 1). Proposition 3.9. Given p ∈ M. there exist a neighborhood V of p in M, a number > 0 and a map γ :(−2, 2)×U → M, where U is defined to be {(q, w) ∈ TU; q ∈ U, w ∈ TqM, |w| < }, such that t → γ(t, q, w), t ∈ (−2, 2), is the unique geodesic of M which starts at q with velocity w < ||. Proof. The geodesic γ(t, q, v) of proposition 3.6 is defined for t ∈ (−δ, δ) and |v| < 1. By lemma 3.8, γ(t, q, δv/2) is defined for t ∈ (−2, 2), with δv δ1 δ1 | 2 | < 2 . Thus if we take = 2 , we obtain that γ(t, q, w) is defined for t ∈ (−2, 2) and |w| < . In the next lectures we will use this proposition to define exponential map as exp(q, v) = γ(1, q, v). Then use this to prove minimizing properties of geodesic.