Differentiable Manifolds

Points in a Euclidean space are written as a linear combination of basis vectors. These are almost always the coordinate directions. For example, we have used this notation here to describe points in the as (x,y,z). For flat spaces this works well, but for mapped regions there are two coordinate systems available, one in the base space (which becomes a basis for the tangent space), and one in the embedding space. It is often convenient to work in the base space, for example, when writing Newton's laws of motion for a marble rolling in a bowl.

As an example, consider the surface of a sphere. Since we live on one of these, these concepts should be familiar. We don't refer to places by spatial coordinates (e.g. position in the solar system), but use coordinates local to the surface of the earth. The directions we call "north", "west" and "up" serve as a local coordinate system (although calling "south" "minus north" probably won't impress your friends at a party). These directions, "north", "west" and "up" are local because they change depending on where you are on the earth. "north" and "west" are a basis for the tangent plane to the earth, and "up" is the the normal. These directions can be constructed by making a small displacement in latitude, longitude, and radius repectively and using the resulting spatial displacement.

This idea generalizes to differentiable manifolds. A "differential structure" on a manifold is a local description of the tangent space and normal space. It includes bases for each of these spaces, and first derivatives of the bases. This information is needed to write differential equations on the manifold.

The "usual" way to put a differential structure on a manifold is to use a basis for the tangent plane which comes from derivatives of the chart mappings. For example, a two dimensional manifold (base space dimension=2) in a three dimensional space (target space dimension=3) with chart

(x(r,s),y(r,s),z(r,s))

would get a basis for the tangent plane at (r,s) consisting of the two vectors

    e_r=(dx/dr,dy/dr,dz/dr)/sqrt((dx/dr)^2+...) 
and 
    e_s=(dx/ds,dy/ds,dz/ds)/sqrt((dx/ds)^2+...).

Adding the additional vector which is normal to both of these gives us a complete basis for points in the target space.

This situation is a little different from Euclidean spaces, where the basis is constant. In Eucildean 2-space, the charts are the identity, so that (x(r,s),y(r,s))=(r,s) So, the basis for the tangent space is

    e_r=(1.,0.)
and 
    e_s=(0.,1.)

On manifolds where the basis for the tangent space is not constant, derivatives of a point which is described by it's coordinates must take into account the way that the basis vectors change, in addition to how the coordinates change. For example,

  A=(x,y,z) -> A = x e_1 + y e_2 + z e_3

  dA/ds = (x_s,y_s,z_s)+ x de_1/ds + y de_2/ds + z de_3/ds

This is what gives rise to the Coriolis force when Newton's law is written for points on the sphere.

The process of writing differential equations on manifolds is simpler if the expressions for the basis are not used directly. If you have tried to write Newton's law on a sphere you may have noticed this. What is really needed is to know the inner products of the basis vectors (called the metric g_{ij}), and the derivatives of the basis vectors (the Christoffel symbols, or the connection G_{ij}^k). These are defined to be

 g_{ij}(r)=e_i(r).e_j(r)

 G_{ij}^k(r)= d e_i(r)/dr_j . e_k(r)

If we define a velocity vector, which is in the tangent space at each point on a 2-manifold, it can be written as

(v_r,v_s)=v_r e_r(r,s)+v_r e_s(r,s)

The Euclidean norm of the velocity is

|v|^2 = v_r^2 g_{rr} + 2 v_r v_s g_{rs} + v_s^2 g_{ss}

If we travel along a path (r(t),s(t)) on the manifold, our velocity is a vector in the tangent plane,

v(r(t),s(t))=(r',s'),

and the acceleration is the derivative of the velocity

 dv/dt =  (dv_r/dt,dv_s/dt) + v_r de_r/dt + v_s de_s/dt
       = (r'',s'')+ r'(G_{rr}^r r'+G_{rs}^r s')+s'(G_{rs}^s r'+G_{ss}^s s')

So Newton's law F=mA picks up some extra terms, involving the connection and metric. These terms are associated with the manifold, and can be used to write other differential operators, like the Gradient and Laplacian, in a coordinate free form.

A Manifold need not be differential. The member function isitDifferential() returns TRUE if a differential structure is present. Manifolds which are Differential provide the following member functions

getTangentSpaceDimension() -- returns the dimension of the tangent space.
getTangentSpace() -- returns a basis vector for the tangent space.
getNormalSpaceDimension() -- returns the dimension of the normal space.
getNormalSpace() -- returns a basis vector for the normal space.
getDistance() -- returns the distance between two points.
getMetric() -- returns an element of the metric.
getConnection() -- returns an element of the connection.
getGeodesicTanget() -- returns the tangent to the geodesic between two points.
isGeodesicConstant() -- answers the question whether the geodesic is a straight line.
projectOntoManifold() -- projects a point in the target space onto the manifold.

There are formulas for these quantites in terms of the charts of the manifold. However, the metric and connection can be defined independantly of the charts. For example, a map of North America (a "flat" map) has one chart with the identity mapping. The fact that the distance scale changes as you go from Mexico to Canada can be reflected in the metric, even though the basis for the tangent space does not change.

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