1.Understands the Concept of Geodesic

Differential Geometry part 9

GEODESICS

objectives

1.Understands the concept of Geodesic

2. Familiarize the concept through examples

introduction

In plane geometry, the straight lines play a very important role as the basis for most constructions and for the formation of most figures studied. However, there are several properties of straight lines and it is not clear which is the most important. That is, it is not clear which of these properties should be taken as the definition of a straight line on an arbitrary sufrace. For example:

SL1 Straight lines have (plane) curvature zero.

SL2 Straight lines give the shortest path between two points.

SL3 Given two points there is a unique straight line joining them.

SL4 The tangent vectors to straight line are all parallel.

We shall generalize each of these properties to curves on surfaces and explore their interrelationships.

Definition A geodesic on a surface M is a unit speed curve on M with geodesic curvature equal to zero everywhere.

This is an immediate translation of property SL1 above. Since the geodesic curvature of a plane curve is its plane curvature, this means that straight lines in the plane are geodesics.

Proposition A unit speed curve in M is a geodesic if and only if

Proof.

The proof the proposition is so clear that we know is the geodesic curvature.

Proposition Let be a unit speed curve, x a coordinate patch, and write is a geodesic if and only

…(1)

for

Proof.

We know that

Since and are independent, the result follows.

Proposition A unit speed curve on a surface M is a geodesic if only if is everywhere normal to the surface (i.e., is a multiple of the normal to M).

Proof:

is normal to the surface if and only if

Example1. Consider the great circle on Then is the inward pointing normal to at is a geodesic since a normal to at is Since there is nothing geometrically special about this particular great circle.

Proposition. Let M be a surface of revolution generated by the unit speed curve Then

(a) every meridian is a geodesic; and

(b) a circle of latitude is a geodesic if and only if the tangent to the meridians is parallel to the axis of revolution at all points on the circle of latitude.

Proof. M may be parameterized by

Since t is arc length along the generating curve, the metric matrix is,

Let so we can compute the Christoffel symbols as follows:

and all other are zero. A unit speed curve is a geodesic if and only if it is a solution of the differential equation

…(2)

…(3)

(a) A meridian is given by constant. Then and are zero and (3) is satisfied. Along a meridian so that and Thus (2) is satisfied. Hence each rneridian is a geodesic.

(b) A circle of latitude is given by constant. Then Since has unit speed,

Thus we have

This last equation implies that (Note r is constant if t is.) Thus and a circle of latitude satisfies Equation (2) if and only if This occurs if and only if is parallel to the axis of rotation (0, 0, 1).

Example2. Let M be the Monge patch Let be the (non-unit speed) curve in M. Now

and

so that

and

so that Hence and is a geodesic. Hence the proof.

In the plane, a straight line is determined once a point on the line and the direction of that line (at that point) are given. The next result says that this is true for geodesics in general.

Theorem1. Let P be a point on a surface M and let X be a unit tangent vector at P . Then, if is given, there exists a unique geodesic with

Proof: Let x be a patch about P with and If there is a geodesic then we must have with initial conditions Conversely, any solution to this initial value problem is a geodesic with the required properties, if the solution is a unit speed curve. Picard's theorem implies there is a unique solution to this initial value problem, for values of s near We must show that the solution to the differential equation is unit speed.

Let Then

which is zero since solves the differential equation. Thus is constant. Since must be identically l. Thus is a unit vector and is a unit speed curve, hence in this case a geodesic.

Repeated application of this gives defined for s in some open interval (a, b), even though the image of may not be all in one coordinate patch. The proof shows that is unique at all points where it is defined.

Example3 Let M be the surface consisting of all the points in except (2, 0). Let P = (0, 0) and let X = (1, 0) be the unit vector pointing in the direction of the positive x-axis. The associated geodesic is and is defined for all Since is continuous, it is impossible to define as anything but (2, 0), which is not in M. It is therefore possible for the geodesic, which is the solution ol the differential equation, to be defined only for

Remark The question as to whether a geodesic extends indefinitely, i.e., is defined for all is metrical. One calls a surface complete if every geodesic extends indefinitely. A famous theorem due to Hopf and Rinow says that M is complete if and only if it is complete as a metric space.

The next theorem tells us that geodesics also possess property SL2 of straight lines. The proof we give is a variant of Gauss's proof. The technique of the proof is to start with a length-minimizing curve and assume its geodesic curvature is not zero. We "wiggle" the curve to form a family of curves with the same end points as and with The function which gives the length of must have a minimum at so that This fact together with integration by parts yields a contradiction. The idea of differentiating with respect to the "wiggle" parameter t is due to Gauss. A proof like this is called "variational" and is a cornerstone in the calculus of variations.

Theorem2. Let be a unit speed curve in a surface M between points and . If is the shortest curve between P and Q, then is a geodesic.

Proof. Let and be the geodesic curvature of . We shall show that Suppose Then there exist numbers c and d with on and the image of under contained in a coordinate patch x. Note that the segment of from to must be the shortest curve joining and or else there is a piecewise regular curve from to to to that is shorter than . But is assumed to give the shortest curve from to .

Let be a function defined for such that

(If is then will work. ) If then in the patch x we have for some Let be given by and define a family of curves by

where is small enough. is a curve from to for each choice of with We may also write

The length of is has a minimum for since gives the shortest path.

At

Thus

But was constructed so that Thus

This contradiction implies

The above theorem is true if is only assumed to be piecewise regular unit speed The converse is false. A geodesic need not minimize distances. Let P and Q be two points on with There are

two geodesics of different lengths joining P to Q, corresponding to the two arcs of the great circle through P and Q.The longer geodesic does not minimize length.

The next two examples show that, in general, property SL3 of straight lines is false for geodesics.

Example4. Let M be the surface of example 2. Then there is no geodesic joining the points (0, 0) and (4, 0).

Example5. Let M be the unit sphere with P and Q the two poles. There are an infinite number of great circles through the poles. Hence there is not a unique geodesic frorn P to Q.

The closest we can come to having property SL3 is the following theorem

which is local in nature (i.e., it says nothing about geodesics between "distant"

points).

Theorem3 Let P be a point on a surface M .Then there is an

Neighborhood u of p such that any two points of u can be joined by a unique geodesic of shortest length ,and this geodesic is contained in u

Definition A vector field along a curve is a function X which assigns to each a tangent vector X(t) to M at .

A differentiable vector field X(T)along is parallel along is is perpendicular to M

Example6 Let be a plane curve Let X(t)=(A(t),B(t),0) be a vector field along . =.The normal to the surface is (0,0,1). Hence is perpendicular to the surface if and only if =0= .therefore X is parallel along if and only if A and B are constants. This is a typical and in fact characterizes the plane locally.

SUMMARY

Now let us summaries the session here we begin with the various definitions of straight lines .After that we come across the concept of Geodesic and its properties .Through various examples we discussed the main features of Geodesic

ASSIGNMENT

1.Show that a meridian of a surface of revolution is a Geodesic without solving the differential equations .

2. Let M be the surface given by x2 +y2 –z2 =1.Find as many Geodesic as you can

3.Let M be a surface and a plane that intersects in a curve Show that is a Geodesic if is a plane of symmetry of M that is the two sides are mirror images

FAQ

1.Every great circle of is a geodesic. Is the converse true

Answer:

The converse is true: if is a geodesic of then is an arc of a great circle.

QUIZ

1. A unit speed curve in M is a geodesic if

(a) (b) (c) (d) none of the above

2. Let be a straight line in a surface M then is a

(a) curvature (b) geodesic (c) tangent plane (d)

None of these

answers

1.(a)

2.(b) geodesic

GLOSSARY

Geodesic : A geodesic on a surface M is a unit speed curve on M with geodesic curvature equal to zero everywhere.

Vector field along a curve: A vector field along a curve is a function X which assigns to each a tangent vector X(t) to M at .

REFERENCE

1.J.A.THORPE,Introduction to Differential Geometry

2.B.O.NIELL,Elementary Differential Geometry

3.S.Sternberg,Lecture notes on Differential Geometry

4..Do carom, Differential Geometry of curves and surfaces

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