Geodesic

Motivation (informal definitions)

First, given a surface and two points $A, B$ on it, we can conceive a geodesic like the (in some sense locally) shortest curve joining $A$ and $B$ .
But there is other underlying idea behind geodesics. They are the straightest lines inside the surface. That is, they are the curves with less curvature: the curve necessarily have the normal curvature provided by the surface so we only can control the geodesic curvature. These curves appears to be straight lines to the inhabitants of the surface, in the sense that their geodesic curvature is 0.

These two motivations are related. Following @needham2021visual page 118, if the curve is the shortest posible joining any two points on it, the normal vector, which in some sense can be "feeling" like the "restorative force" of the curve, must point perpendicularly to the surface, since if not the curve wouldn't be the shortest posible. Therefore, all the curvature of the curve is in the normal curvature component (see normal and geodesic curvature of a curve) and therefore the geodesic curvature is zero.

Possibly related: curvature like a driving force.

Another motivational idea: if the geodesic is traced on the surface of a fruit and then is peeled from the surface and laid flat on the table it becomes a straight line (@needham2021visual pages 13-14).

Definitions

Definition 1 (for immersed manifolds)
Given a pseudo-Riemannian manifold, a geodesic it is a curve such that the geodesic curvature is cero.
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Definition 2 (autoparallelly transported curve)
Consider that we have a manifold $M$ and a covariant derivative operator $\nabla$ . A curve $γ$ is a geodesic if the vector field $v = γ^{'} (t)$ defined along $γ$ is constant along $γ$ , that is,

D_{v} v = 0,

where $D$ is the covariant derivative along a curve. Also known as autoparallelly transported curve.
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Definition 3
Given a curve in a Riemannian manifold or pseudo-Riemannian manifold, it is called a geodesic if it is stationary with respect to the functional length of a curve.
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These definitions are equivalent if $g$ and $\nabla$ are compatible (see Levi-Civita connection).

Definition 4
There is a weaker definition, in which the speed of the geodesic is not constant, only the direction is. In that case, they are defined by

D_{v} v = α v,

for certain function $α$ defined on the image of the curve, $γ ([0, 1])$ . They are called pregeodesics. Also known as autoparallel curves.
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Existence and computation

Proposition
Given a manifold $M$ with a covariant derivative $\nabla$ , let $p \in M$ and $ξ \in T_{p} M$ . There exists a unique geodesic $γ : I \to M$ such that $γ (0) = p$ , $γ^{'} (0) = ξ$ , and it is maximal in the following sense: if there is another curve satisfying the same conditions, its domain is contained in $I$ , and they coincide on the intersection.
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Proof
Existence and uniqueness of parallel transport (see here).
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To compute a geodesic curve in a local chart we use the Christoffel symbols. The geodesic equations are second order equations given by:

\frac{d^{2} x^{i}}{d t^{2}} + Γ_{j k}^{i} \frac{d x^{j}}{d t} \frac{d x^{k}}{d t} = 0

In two dimensions:
Sure, in a 2-dimensional Riemannian manifold, where $i, j, k \in {1, 2}$ , the geodesic equations will be given as follows:

\frac{d^{2} x^{1}}{d t^{2}} + Γ_{11}^{1} \frac{d x^{1}}{d t} \frac{d x^{1}}{d t} + Γ_{12}^{1} \frac{d x^{1}}{d t} \frac{d x^{2}}{d t} + Γ_{21}^{1} \frac{d x^{2}}{d t} \frac{d x^{1}}{d t} + Γ_{22}^{1} \frac{d x^{2}}{d t} \frac{d x^{2}}{d t} = 0

\frac{d^{2} x^{2}}{d t^{2}} + Γ_{11}^{2} \frac{d x^{1}}{d t} \frac{d x^{1}}{d t} + Γ_{12}^{2} \frac{d x^{1}}{d t} \frac{d x^{2}}{d t} + Γ_{21}^{2} \frac{d x^{2}}{d t} \frac{d x^{1}}{d t} + Γ_{22}^{2} \frac{d x^{2}}{d t} \frac{d x^{2}}{d t} = 0

This simplifies further by noting that the Christoffel symbols are symmetric (since the connection as no torsion) in the lower indices, i.e. $Γ_{j k}^{i} = Γ_{k j}^{i}$ , so we have:

\frac{d^{2} x^{1}}{d t^{2}} + Γ_{11}^{1} {(\frac{d x^{1}}{d t})}^{2} + 2 Γ_{12}^{1} \frac{d x^{1}}{d t} \frac{d x^{2}}{d t} + Γ_{22}^{1} {(\frac{d x^{2}}{d t})}^{2} = 0

\frac{d^{2} x^{2}}{d t^{2}} + Γ_{11}^{2} {(\frac{d x^{1}}{d t})}^{2} + 2 Γ_{12}^{2} \frac{d x^{1}}{d t} \frac{d x^{2}}{d t} + Γ_{22}^{2} {(\frac{d x^{2}}{d t})}^{2} = 0

Here $x^{1}$ and $x^{2}$ are the coordinates of the 2-dimensional Riemannian manifold.

Other properties

As said in the "Motivation" section above, geodesics minimise the length of a curve and the energy of a curve. In @malham2016introduction page 81 it is shown, by using Cauchy-Schwarz inequality, that minimising the energy of a curve is the same as minimising the length plus the additional requirement that the speed $∥ c^{'} (t) ∥$ stays constant.
A vector field whose integral curves are geodesics is called a geodesic vector field.
At any point of a geodesic in a surface, the normal of the curve is parallel to the normal of the surface.
In a surface of revolution, the meridians are geodesics.
The geodesics have a "behavior" influenced by the Gaussian curvature. This is reflected in the Jacobi equation.
Their generalization to other dimensions different from 1 is related to totally-geodesic manifolds.
If two linear connections on an open subset $U$ of $R^{2}$ give rise locally to the same families of geodesics up to reparametrization (pregeodesics), they are indeed the same connection. Intuitively, the reason is that a linear connection on a manifold is equivalent to a unique notion of parallel transport along curves in the manifold, and this notion of parallel transport is what determines the geodesics for the connection. The parallel transport can be made by pieces of geodesics, see the intrinsic construction. If two connections have the same geodesics, then they must have the same parallel transport, and thus they must be the same connection. For a computational proof, see contorsion.
On the contrary, two different Riemannian metrics can give rise to the same geodesics locally in a region, but they are not necessarily the same metric. In other words, two different Riemannian metrics can have the same geodesic flow, but they can differ in other ways. Let's take a look at a simple example to illustrate this point. Consider the flat Euclidean metric in $R^{2}$ and a metric that has been scaled by a positive factor. These two metrics will produce the same geodesics, which are straight lines in this case. However, these two metrics are not the same because one is a scaled version of the other. The reason why this is possible is that two metrics can have the same Levi-Civita connection and hence the same geodesics, but differ in other properties like volumes, angles, etc. In more formal terms, this question involves projective equivalence of Riemannian metrics, a topic studied in differential geometry. Two metrics are projectively equivalent if they have the same unparametrized geodesics. There are nontrivial examples of projectively equivalent but distinct Riemannian metrics, even in dimension 2.
In the context of general relativity, geodesics serve to model gravity.

Estimation of geodesics

See xournal 259 and mathematica 020 to see estimation through circle-elipses