geodesics equation in Riemannian manifold

Let $$(M,g)$$ be a Riemannian metric and $$U \subset M$$. We know that in local coordinates on $$U \subset M$$ the equation for a curve to be a geodesic is:

$$0=γ ̈ =(d^2 γ^k)/(dt^2 ) ∂_k+(dγ^i)/dt (dγ^j)/dt Γ_{ij}^k ∂_k$$

Where $$Γ_{ij}^k$$ are the Christoffel symbols. This is a second-order system of differential equations.

Now are the geodesics always smooth in a Riemannian metric? Thanks in advance.

The geodesic equation says that, locally, geodesics are at least $$C^2$$. Since the geodesic equation is invariant under coordinates changes, we find out that a geodesic is at least $$C^2$$ in all of its range. There is no smoothness $$C^\infty$$ requirement here.

• If you start with a smooth metric, rather than a $C^2$ metric? – Ted Shifrin Jul 1 at 1:11
• That doesn't change things, $C^2$ comes from the geodesic equation being a 2nd order system of ODEs – big-lion Jul 1 at 10:58
• That's odd. So the exponential map on a smooth manifold is never better than $C^2$? I think you need to review your ODE theory. – Ted Shifrin Jul 1 at 15:14
• I didn't say that, rather I said the solutions are at least $C^2$. You can't guarantee smoothness, but of course it can be there. – big-lion Jul 1 at 19:27
• Do you have a good reference to this case (when the metric is smooth) – walaa Jul 1 at 20:01

Note: The Einstein summation convention is in force throughout this answer.End of Note.

Our OP walaa's question,

Now are the geodesics always smooth in a Riemannian metric?

appears to be somewhat of a misstatement, insofar as smoothness is technically defined with respect to local coordinate patches of a given atlas; both the smoothness of the metric tensor $$g_{ij}$$ and of the Christoffel symbols $$\Gamma_{ij}^k$$ are then affirmed if these functions are smooth in such coordinates. A more carefully stated question along these lines might read:

Are the geodesics of a smooth Riemannian metric smooth?

This is indeed the case, as may be seen by observing that the coefficients $$\Gamma_{ij}^k$$ occurring in the geodesic equation

$$\ddot \gamma = \dfrac{d^2 \gamma^k}{dt^2} \partial_k + \Gamma_{ij}^k \dfrac{d\gamma^i}{dt} \dfrac{d\gamma^j}{dt} \partial_k = 0 \tag 1$$

are themselves smooth functions on $$M$$, being given in terms of the $$g_{ij}$$ by

$$\Gamma_{ij}^k = \dfrac{1}{2}g^{km}(g_{mi, j} + g_{mj, i} - g_{ij, m}), \tag 2$$

where

$$[g^{ij}] = [g_{ij}]^{-1}; \tag{2.5}$$

in these equations we understand that

$$\partial_k = \dfrac{\partial}{\partial x_k}, \tag 3$$

and

$$g_{ij, m} = \dfrac{\partial g_{ij}}{\partial x_m}, \tag 4$$

and so forth.

We may cast (1) in the familiar first order form by setting

$$\dfrac{d\gamma^i}{dt} = \beta^i, \; 1 \le i \le \dim M; \tag 5$$

then

$$\dfrac{d^2\gamma^i}{dt^2} = \dfrac{d\beta^i}{dt}; \tag 6$$

(1) may then be written

$$\dfrac{d\beta^k}{dt}\partial_k + \Gamma_{ij}^k \beta^i \beta^j \partial_k = 0, \tag 7$$

which corresponds to the collection of $$\dim M$$ first order, non-linear ordinary differential equations

$$\dfrac{d\beta^k}{dt} + \Gamma_{ij}^k \beta^i \beta^j \ = 0, \; 1 \le k \le \dim M, \tag 8$$

that is,

$$\dfrac{d\beta^k}{dt} = -\Gamma_{ij}^k \beta^i \beta^j \ = 0, \; 1 \le k \le \dim M; \tag 9$$

since the $$\Gamma_{ij}^k$$ are functions of the position coordinates $$x = (x_1, x_2, \ldots, x_{\dim M})$$ in $$M$$, along $$\gamma(t)$$ we have

$$\Gamma_{ij}^k(x) = \Gamma_{ij}^k(\gamma(t)) = \Gamma_{ij}^k(\gamma^1(t), \gamma^2(t), \ldots, \gamma^{\dim M}(t));\tag{10}$$

(9) then becomes

$$\dfrac{d\beta^k}{dt} = - \Gamma_{ij}^k(\gamma^1(t), \gamma^2(t), \ldots, \gamma^{\dim M}(t))\beta^i \beta^j = 0, \; 1 \le k \le \dim M; \tag{11}$$

these together with (5) form a system of $$2\dim M$$ ordinary, non-linear differential equations for the $$\gamma^i(t)$$, $$\beta^i(t)$$, $$1 \le i \le \dim M$$.

We observe that if the functions $$\Gamma_{ij}^k(\gamma^1, \gamma^2, \ldots, \gamma^{\dim M})\beta^i \beta^j$$ occurring on the right-hand side of (11) are in fact $$C^m$$ in the $$\gamma^i$$, $$\beta^i$$, then (5), (11) forms a $$C^m$$ system of non-linear, ordinary differential equations for the $$\gamma^i$$, $$\beta^i$$. It then follows from the standard theorems that the solution functions $$\gamma^i(t)$$, $$\beta^i(t)$$, $$1 \le i \le \dim M$$ are themselves at least $$C^m$$; since the $$\Gamma_{ij}^k$$ in fact depend on the first derivatives of the $$g_{ij}$$ (cf. (2)), we may infer that for $$C^m$$-smooth $$g_{ij}$$ the $$\Gamma_{ij}^k$$ are $$C^{m - 1}$$-smooth, and hence so the $$\gamma_i(t)$$, $$\beta_i(t)$$ are at least $$C^{m - 1}$$. Since these assertions bind for any $$m \ge 1$$ it follows that if the metric $$g_{ij}$$ is $$C^\infty$$, so are the functions $$\gamma_i(t)$$, $$\beta_i(t)$$, $$1 \le i \le \dim M$$. Thus the geodesics of a smooth Riemannian metric are themselves smooth.

• Thank you this what i want but i also need from which refrence this information – walaa Jul 5 at 17:17
• @walaa: thanks for the "acceptance"; as far as references go, not sure of any off the top of my head, but you might check out Milnor's Morse Theory and/or Kobayashi and Nomizu's Foundations of Differential Geometry, vols. I & II., – Robert Lewis Jul 5 at 17:20