Sierpinski (<1K) The Coordinate based approach
    By defining an M-curve by means of orientation Ip and projector ¯ we have presented the standard tensors of manifold calculus and verified the vanishing point-divergence of the Einstein tensor all without refernce to coordinates other than to simplify the occassional proof.
    We will now run through the standard noncovariant coordinate coordinate representations of these entities since from a programmers' perspective these may be relevant. We will touch on the somewhat arcane metric-focussed covariant coordinate representations in the section on General Relativity.

    M-field Ip can be expressed in UN via NCM scalar fields = eij...m¿Ip though the normalisation condition renders one such field redundnat apart from sign. In the mapspace ¦Ñ-1(Ip) is everywhere the map pseudoscalar.
    In UN the projector 1-tensor ¯ can be expressed with regard to a universal basis as N2 scalar fields although with regard to an extended tangential frame at p it can be regarded as an identity matrix having the last N-M terms in the lead diagonal zeroed.

    The Riemann curvature tensor cpaÙb, as a bivector valued function of bivectors, constarined within and upon Ip requires MC22 = ¼M2(M-1)2 scalar fields in theory although the Bianchi symmetries reduce the generality considerably. At a fixed given p Î CM,  mapping bivectors in Ip to bivectors in Ip induces the obvious Cx(aÙb) = ¦-Ñp(cp¦Ñp(aÙb)) 2-tensor acting in the mapspace which we can represent with ¼M2(M-1)2 scalar fields Cklij º ekl¿Cx(eij)

    The alternate view of the Riemann curvature as a bivector specific 1-tensor , or a (1;3)-tensor antisymmetric in two of its arguments, provides the alternate coordinate form Ckijl º ek¿(Cx(eij)×el)

     cpaÙb and Cx(aÙb) are alternate representations of the same "thing", although cp defined over subspace Ip of UN can more readily be "extended" out of CM.     We can contract the curvature tensor to obtain the Ricci 1-tensor)
    cpb º Ña¿cpaÙb = åi=14 ei¿cpeiÙb = åi=14 ei¿ (( Я(b)wei) - ( Я(ei)wb) + (wb×wei))    , a point-dependant symmetric 1-vector-valued function of 1-vectors . Note that the divergence is with respect to direction rather than a point and is computable entirely "at" x.
    The mapspace representation of the Ricci 1-tensor consists of M 2 scalars Ckj º ek¿ Cx(ei)   = åj=1M Ckjij or with Cij º åk=1M Ckijk which itself contracts to the scalar curvature R = åi=14 Cii .
    The mapspace Einstein tensor is given by M2 scalar fields   Gkj º Ckj - ½R .

Streamline Coordinates

The following is a more traditional approach to manifold calculus which we include for completeness. This section can safely be skipped by readers interested only in multivector methods.

    Suppose we have a 1-field vp defined over M-curve CM such that vp Î Ip . It follows from a mathematical result concerning the unique solvability of ordinary differential equations that, provided vp is sufficiently smooth and is nowhere 0-valued, for every p0ÎCM there is a unique "tangent matcher" path in UN for vp :
    Ci p = { p(t) : t Î [t0,t1] }  [ where t0<0<t1 ] with p(0)=p0 ; p(t) Î CM and  (/t)p(t) = vp(t) " t Î [t0,t1] . ( p(0)=p0 ensures that the parameterisation is unique up to choice of domain range [t0,t1])
    Such paths are called integral curves or streamlines for vp. We will use both terms interchangeably here. Streamlines cannot intercept eachother (other than at points with vp=0 which we have disallowed) . Since every pÎCM has such a curve, the set of all integral curves for each vp (known as a congruence for vp) "fills" and effectively defines CM.
    It is important to note that the 1-curve streamlines of field vp are independant of the magnitude of vp, (given vp2 ¹ 0) since it is the unit tangent that defines a 1-curve.

    Suppose now we have M not-necessarily mutually orthogonal unit 1-fields ui p defined over M-curve CM satisfying
    u1(p)Ùu2(p)Ù..uM(p) = apIp     " p Î CM where scalar field ap > 0 " p .
    An obvious example is ui p=¦Ñx(ei)= hip  where CM=¦(Map) is a point embedding specification for CM.

    Given a general multivector-valued function F(p) defined over CM , ui p generates at each pÎCM a specific "streamlined" derivative of F which is the tangential derivative within the 1-curve ui p streamline. We will write this as
    dli(F(p)) º  F(pi(li))/li . We will sometime ommit the brackets and write dliF(p).
    If we so differentiate the identity map F(p)=1(p)=p we recover ui p and, indeed, one can think of 1-field ui p as being the streamlined derivative operator field dlip . We will accordingly use   the symbol Я(ui p) interchangeably with dli .

    We introduce the notation  Dli¬i p to indicate the point pi(Dli) Î CM reached by moving from p by arclength Dli along the li (ie. the ui p) streamline through p.
     Dli¬i p º òCi p [0,Dli] 1 dp = ò0Dli pi'(li) dli
    We can extend  Dli¬i  to act on general multivector fields by  Dli¬i F(p) º F( Dli¬i p) .
    This defintion is integrative, good for all Dli within the neighbourhood of ¦ applicability. For small Dli=e we can apply Taylor's Theorem to the streamline li-parameterisation and obtain F( e¬i p) = eedliF(p) and in particular [  Set F(p)=p ] we have  e¬i p = eedli(p) which we can write as
     e¬i  = eedli .

    We might be tempted to try to parameterise CM using y:Map®CM defined by
    y(Dl1, Dl2, ..Dlm, q0) º  Dlm¬M  Dlm-1¬M-1 ... Dl2¬ Dl1¬q0 =( )=  Dlm¬M ( Dlm-1¬M-1 (... Dl2¬( Dl1¬q0))..) .
    [   That is, starting from a fixed reference point q0ÎCM we follow the v1(p) stream line for arclength Dl1 to a new point  Dl1¬q0 from which we follow the v2(p) streamline for arc length Dl2 to  Dl2¬ Dl1¬p0 and so forth. After M such steps we arrive a final point qMÎCM which we can associate with parameters (Dl1,..,Dlm). ]
    Any M scalars (sufficiently small that we stay within the neighbourhood) interpreted in this way define a point in CM, but not necessarily uniquely. If it is possible to reach the same point by more than one such route, then the mapping is noninvertible and does not provide a true coordinate system for CM.
    Only if the dli (and hence the  Dli¬i ) everywhere commute with eachother do the li provide a true coordinate system.
[ Proof : ...  .]

    ( e¬i  ×  e¬j ) is a 1-field defined over CM returning a UN 1-vector ( e¬i  ×  e¬j )(p) = (eedli × eedlj)(p) = ... = e2(dli×dlj)(p)   + O(e3) .
    Taking e®0 we can visualise  (dli×dlj)(p) º [ ui p,ujp ](p) as the 1-vector difference at p between following i and then j as opposed to j and then i streamlines by small arclength e.

Lie Derivative
    We make very little use of Lie derivatives in this work. They are described here only for completeness. This section can safely be skipped by readers interested only in multivector methods.

Lie Bracket

    The Lie Bracket is a highly general mathematical construct. With regard to 1-fields in an M-curve we define it by
    [ui p,ujp](v) º  dli(dlj[v]) -  dlj(dli[v]) = 2(dli×dlj)v
which we can write as [ui p,ujp] = dliujp - dljui p .
    The use of × above is semantically legitimate if we extend the definition of commutator multivector product × to operators in the obvious appropriate way.

Lie Drag

     Any one of our ui p generates a congruence for CM so let us fix on ui p as a "preferred" congruence with associated dervivative field dli . Consider two points p0 and p0 in CM with p0= Dli¬i p0 for small Dli>0. Clearly p0= -Dli¬i p0 .
    For j¹i, the specific ujp streamline through p0 (i¹1) at p with associated directed derivative dlj generates a distinct path pj*(lj) =  -Dli¬i pj(lj) passing through point p0 =  -Dli¬i p0 known as the Lie drag of ujp . [  Here * indicates "new" or "modifed" rather than a multivector dual ] . This path need not be the streamline at p0 of any of our uk p 1-fields and, in particular, it need not be the lj streamline there .
    It is  however, a streamline having at p0 a tangent vector written (iÝ-Dli(ujp))(p0) fully within Ip0 and an associated streamlined differential operator dlj* = (iÝ-Dli(dlj))(p0)  .
    Though dlj* is not one of the dlk, it is not general. Because of its construction it commutes with dli (ie. dlj*(dli(F))) = dli(dlj*(F))) . [ Proof : ....  .]
    We refer to     (iÝ-Dli(ujp))(p0)     ( or (iÝ-Dli(dlj))(p0) ) as the Lie drag of ujp ( or dlj ) from p0 to p0=  -Dli¬i p0 .
    We define the Lie drag of a scalar field f(p) defined over CM by (iÝ-Dli(f))(p) º f( -Dli¬i p) so that (iÝe(f))(p) = eedlif .                                              7

Lie Derivative
    The 1-vector Lie derivative of a scalar field is simply dlif(p) = dlif(p) .

    Since the Lie drag of ujp (or dlj) from p0 to p0 can be meaningfully subtracted from ui p (or dli) at p0 . The Lie dervivative at p0 of ujp with regard to 1-field ui p is an operator defined (by its action on a general 1-field F) as
    ( dli(dlj))F(p) º LimDli ® 0 [ ( (iÝ-Dli(dlj))(p0)(F(p)) - dlj(F(p)) )  / Dli ] ïp0 = LimDli ® 0 [2(dli×dlj)F  ïp0 ]
[ Proof : dlj*F ïp0 = dlj*F ïp0 - Dli(dli(dlj*(F))))ïp0 + O(Dli2) = dlj*F ïp0 + Dli(dli(dlj(F))ïp0 + O(Dli2) - Dli(dlj*(dli(F))ïp0 + O(Dli2)
    Hence LimDli ® 0 [ ( (iÝ-Dli(dlj))(p0) - dlj )F  / Dli ] ïp0 = LimDli ® 0 [((dlj* - dlj )F  ïp0 ) / Dli ] = LimDli ® 0 [(dli(dlj(F)) - dlj*(dli(F)) ) ïp0 ] which provides the result assuming dlj* can be safely replaced in the limit by dlj  .]

    Omitting the brackets , we thus have the 1-vector operator identity
    ui pujp = dlidlj = 2 dli×dlj = [ ui p,ujp ] .

Covariant Frame
    Let us now identify the spaces UN and VN and adopt fixed frames {ei} and {ei} in both. With regard to ¿ within VN , {ei} is not a reciprocal frame for {ei}. Rather, we need the point-dependant (nonorthormal) covariant frame {e i º gx-1(ei)} which itself has ¿ reciprocal frame {ei º gx(ei)}
[ Proof :   e i¿ej = gx(gx-1(ei))ej = ei¿ej     Also e i¿ej = gx-1(ei)¿gx(ej) = gx(gx-1(ei))¿ej = ei¿ej  .]
    For brevity we ommit an x subscript from ei , e i , and ¿ . The underline serves to remind us of the point-dependance.


    Any "inner product"  ¿ is a (0;2)-tensor (a potentially point-dependant bilinear scalar-valued function of two 1-vectors in VM) . For any 1-field ( (1;0)-tensor) vp , ¿ induces a scalar-valued directional function ( a (0;1)-tensor)     vp(a) º a¿vp     known as a covector field. We can recover vp at a given p from its covector as vp = Ñavp(a) . A vector and its covector are essentially alternate representations of the same "thing". If we "feed" vp "to" its covector we get "squared magnitude" 0-tensor vp(vp) = vp¿vp
    For ¿ in ÂN , if 1-vector vp is regarded as a 1×N matrix "column vector" , then its covector is the N×1 matrix "row vector" (or transpose) vpT . For ¿ in Âp,q , we take the transpose of the conjugate vp .
    With regard to ¿, we will here define the 1-covector of vp = åi=1N  viei as the 1-vector vpc º  åi=1N viei so that vpc ¿+ a =    vp¿a .
    In a Euclidean space, vpc = vp .
    (1;0)-tensor vp has "contravariant" coordinate representation åiviei where vi º ei¿vp .
    (0;1)-tensor vp has "covariant" coordinate representation åivie i where vi º vp(ei) = ei¿vp and e i is the 1-covector of ei which we associate with the 1-vector e i.

Parallel transport

    Suppose we were to drag heavy iron girders around the (assumed perfectly spherical) Earth for a bet. We might start at the North Pole, with two girders girder (assumed straight and with ends clearly distinguished) lying "horizontal" on the ground ; perpendicularto a "vertical" flagpole we assume to mark the Pole and parallel to eachother.
    Since the globe is spherical, the girder is actually balanced tangentially on a single point but the radius is so vast that the ground is for all girder-related purposes locally flat. Both gripping one end we start to drag the girder along a meridian, always pulling due South, that is, allowing no rotation in the local "horizontal" tangent plane.  When we reach the equator, or girder has retained its length and is still facing South (local coordinates) but, speaking 3-dimensionally, it is now parallel to the flagpole we leftbehind. If we then push the second girder sideways from the North Pole without rotation the girder maintains an East-West alignment. Once we reach the equator (at a different point to previously) we then drag the girder "lengthwards" ¼ of the way round the equator to join the first girder, to which it is now perpendicular.

    Moving a 1-vector along a curve while keeping it "held within" an M-curve in this way is known as parallel transport in the M-curve. The example shows that it preserves length but not direction. If we parallel transport a 1-vector from p to q within an M-curve then the result is dependant on the path taken.
    Parallel transport is invoked to provide a notion of "parallelism" for 1-vectors in different tangent spaces (ie. 1-vectors "at" different points of an M-curve) that does not require a higher-dimensional embedding space. The "distance" between two points on a manifold can be meaningfully defined as the arclength of a minimial-arclength path joining the points (a geodesic), but to compare distant directions we need for every qÎCM a function ­qp C(a) Î Ip giving the tangent vector at pÎCM "parallel" to aÎIq at q ÎCM ; as obtained by "parallel transportation" of a "along" a given path C from q to p (eg. along a minimal length geodesic).

    Let i­Dlip0(a) denote the parallel transport of a from p0 for arclength Dli along the ui p streamline.
    We obviously want i­Dlip0(a)ÙI Dli¬i p0 = 0 and we might therefore think of defining i­Dlip(a) = ¯I Dli¬i p0(ujp)~ whenever ¯I Dli¬i p0(ujp) is nonnull but instead we simply postulate a 1-vector i­Dlip(a) within I Dli¬i p0 that satifies i­Dlip(a)2 = a2 .   We will henceforth assume a to be a unit vector.

    The normalisation condition for i­Dlip(a) means that the right-connection ba,p(Dli) º a-1i­Dlip(a) for i­Dli at p satisfies ba,pba,p§ = 1 and is thus a 1-rotor expressable as a 2-spinor ewui p a p(Dli) , where wui p a p(Dli) = cos-1(a-1¿i­Dlip(a) (a-1Ùi­Dlip(a)~ is a UN 2-blade. [  we assume Dli is sufficiently small and CM sufficiently smooth that the subtended angle is comfortably below ½p ]  

    We would like the angle subtended by two 1-vectors parallelly transported along the same streamline to remain the same, ie. i­ep(a) ¿ i­ep(b) = a¿b and we can acheive this by insisting that ui p¿wui p a p(e) = O(e2) .
    i­ep(a) º aewui p a p(e) = a + a¿wui p a p(e) + O(e2)
where for given scalar e, wui p a p(e) is a pure UN-bivector 2-field defined at every pÎCM for all aÎIp .
    Parallel displacement along the ith streamline thus preserves orthonormality and corresponds to a Lorentz rotation having UN rotor Rp i(e) which we assume to tend to a well defined limit as e®0. M such rotor fields fully define parallel transport and effectively define CM.

Linear Connection

    gp(ujp,dli,e) º ujp - i­ep(ujp) = -ujp¿wui p ujp p(e)  + O(e2)
      is a 1-vector lying in general neither in Ip nor I e¬i p. It cannot be properly regarded as a tangent 1-field of the M-curve (traditionally speaking, it is not a "tensor").
    We make the continuity assumption that Lime ® 0[ e¬i ujp - i­-e e¬i p( e¬i ujp)] = -gp(ujp,ui p). [  We use the symbol g here in accordance with traditional GR use of G for the gravitational connection acting in the mapspace. Conflict with the widespread use in multivector literature of g for a (typically Â1,3) fiducial frame is unfortunate. ]
    Taking gp(ujp,ui p) º Lime ® 0[e-1gj p(ui p,e) ] we obtain the connection, the 1-vector component of dli ujp ïp that exists outside Ip.
    gp(ujp,ui p) = -ujp¿wui p ujp = wui p ujp ×ujp .
    gp(ujp,ui p) is orthogonal to both ui p and ujp at p .

    A linear connection (aka. affine connection) derives from an assumed bilinearity of gp which can then can be fully defined by M2 1-fields   gp( hjp , hip ) which then effectively define ­qp C(a). Indeed, some authors define i­Dli as the consequence of a given linear connection.
    We say the connection is symmetric if gp(ui p,ujp) = gp(ujp,ui p) .

    Suppose now that the ui p are everywhere mutually orthogonal so that {ui p} provides an orthonormal frame for Ip with inverse frame {uip}. Because wui p ujp is a 2-blade perpendicular to Ip, we have wui p ujp ×ui p = 0     i¹j . Thus we can define M pure bivectors wui p p º åj=1M wui p ujp which satisfy gp(ujp,ui p) =  wui p p×ujp
    The connection is symmetric if ujp¿wui p p = ui p¿wujp p     " 1£i,j£M .
    A linear connection arises if wui p a is linear in a.
[ Proof : Since the ui p are orthogonal gp((åj=1M bjujp),(åi=1M aiui p)) = (åj=1M bjujp)¿wui p åi 1M= aiui p)}
= (åj=1M bjujp)¿(åi=1M aiwui p ui p ) = åi=1M åj=1M aibj(ujp¿wui p ui p ) = åi=1M åj=1M aibjgp(ujp,ui p)  .]

Within the mapspace
    Parallel transport in the mapspace i­ex(b) º ¦Ñx+eei-1( i­e¦(x)(¦Ñp(b)) ) preserves ¿ instead of ¿.
[ Proof : i­ep(a) ¿x+eei i­ep(b) = i­ep(¦Ñx(a))¿i­ep(¦Ñx(b)) = ¦Ñp(a)¿¦Ñx(b) = a¿xb  .]

    Let b be a map direction at xÎMap with corresponding CM direction a=¦Ñx(b) at p=¦(x). It may not make any geometrical "sense" to subtract a map 1-vector "at" x+eei from one "at" x but we can still do it!
    Gx(b,ei,e) º b - i­ex(b) = b - ¦Ñx+eei-1(i­ep(¦Ñx(b))) = b - ¦Ñx+eei-1(¦Ñp(b) - gp(¦Ñp(b),dli,e))
is, for given ei and scalar e,  a map-1-vector-valued function of map-1-vector b, linear in b if the CM connection gp is linear. We then have Gx(ei,ej,e) = åk=1M  Gkijx(e)ek . Letting   Gx(b,ei) =   Lime ® 0(e-1Gx(b,ei,e)) we achieve an affine connection for the mapspace, symmetric if g is.
    Gx(a,b) = ¦-Ñx(¦Ñ2x(a,b))
[ Proof :   See General Relativity Chapter  .]
    We can think of G as a rule for moving map 1-vectors around. In the chaper on General Relativity we derive an expression for G in terms of the metric.

Directed Coderivative

    The directed coderivative of b(p) with respect to streamline ui p at p is defined by parallel transporting b( e¬i p) back from  e¬i p to p and there subtracting b and dividing by e, in the limit as e®0. It is thus a 1-vector within Ip.
    Ðui p(b) = dlib + gp(b,dli) = dlib + wui p ujp p×b
[ Proof : Ðui p(b(p)) º Lime ® 0[ i­-e e¬i p(b( e¬i p)) - vj(p) / e ]
    = Lime ® 0[ ( i­-e e¬i p(b( e¬i p)) - b( e¬i p)) + ( b( e¬i p) - b(p)) / e ] = gp(b,dli) +dliujp  .]

    In the limit, since li is  the natural (proper) parameterisation, we can consider dli to be equivalent to the N-D directional derivative Ðui p
    Accordingly we can define a directed coderivative operator
    Ða º Ða + wa×
for a given direction aÎIp which when applied at p to a 1-field over and within CM returns a 1-vector in Ip.
    We then have
    i­Dlip(ujp) = eDliÐli e¬i ujp = eDliÐli eDlidli ujp .
[ Proof : i­Dlip(ujp) = vj( Dli¬p) + DliÐlivj( Dli¬p) + Dli2Ð (ui p)2vj( Dli¬p) + ...  .]

    Ða can be regarded as a differential over and "within" CM, although its component parts Ða and wa× cannot.

    Ða generates a 1-vector codel-operator
    Ñp º åk=1M ukpÐuk p = Ñ[CM] + åk=1M ukp(wuk p×) .

Within the mapspace
    Ða induces a differential operator within M-D parameter space MapÌVM
    Ðað(B(x)) = ¦-Ñp(ЦÑp(a)(b(p)))     where b(p) º B(¦-1(p)) .
    Ðað(vp) = Ða(vp) + G(a,vp)     where G is linear as a consequence of the linearity of ¦-Ñp .
    In particular,
    Gp(ej,ei) º Ðeiðej = åk=1M Gkjiek     where point-dependant scalar Gkji is known as a Christoffel symbol.

    We say a 1-field ui p is geodesic if i­Dlip(ui p) = ui p( e¬i p)     " pÎCM, Dli
where Dli is assumed small enough to remain in the neighbourhood. Geodesic fields are thus invarient under parallel transport and satisfy Ðui pui p = 0.
    The physical interpretation of geodesics are the potential trajectories of "free falling" particles and (assuming a natural streamline parameterisation pi'(li)2 = 1) we can then regard the
    pi"(li) + g(pi'(li),pi'(li)) = 0
derived from Ðui p(ui p) = 0 as relating the "acceleration" of a "particle" to its velocity and the local "geometry" g.

    An alternate definition of a geodesic is the "shortest subcurve between two points". If we fix two points p and q on a geodesic streamline then the arclength from p to q is stationary (minimal) for small changes in the path from p to q that keep it within CM. The equivalence of the two defintions is intuitively reasonable and the proof (ommitted here) arises from fairly straightforward calculus

[Under Construction]

    The traditional curvature of a 2D 1-curve p(t) (ie. a 1-curve confined to a 2-plane ) is defined as
    C(t) º Limd ® 0[ d-1 qÐ(p'(t+d) , p'(t)) ] where t is the natural parameterisation - ie. the instantaneous angular tangent change per unit arclenth. C(t)-1 is known as the raduius of curvature and coresponds to the radius of the circle matching the 1-curve to second order at p(t) . For an N-D 1-curve we have 1-curvature p"(t) which is a 1-vector orthogonal to p'(t) ("normal" to the 1-curve) as a consequence of the constancy of p'(t)2 . Its magnitude C(t) is the curvature of the 1-curve confined in the limit to 2-plane p(t) and corresponds to the reciprocal of the radius of the osculating circle.

2-Curvature   as ortho bi-circle
    For a 3D 2-curve we have tangent unit vectors h1,h2 and normal n=h1×h2  forming the moving trihedron at p. Taking p as the origin, the induced coordinates (x1,x2,x3) satisfy
    x3 = ½ (2x3/(x1)2) x12  + (2x3/x1x2) x1x2 + ½ (2x3/x2) x22
    Rotation about e3 diagnonalises as x3 = ½ (k1(x1)2 + k2(x2)2)     where scalar k1,k2 are known as principle curvatures.
    Their product k1k2 is known as the Gaussian curvature, and their average ½(k1+k2) as the mean curvature. If k1=k2 we have no principle 1-spheres but rather a tangent 2-sphere or 2-plane at umbilical or flat p. The Gaussian curvature vanishes before the mean curvature in general.

    If the surface  is parameterised as p(l1,l2) with h1p  = p/l1 , h2p  = p/l2 then
    2p/lilj º hip /lj = åk=12 Gkij hkp  + bij h3p      i,j,k Î {1,2}
    h3p /li = - åk l=12 gklbli hkp      i Î {1,2}

    2-curvature is geometrically representable as ortho bi-circle S1+S2 for 3-blade osculating at p principle 1-spheres S1 and S2. Their product S2S1 embodies the rotation taking S1 into S2, inflecting computationally robustly through a 1-plane if p is a saddlepoint of the 2-curve with negative Gaussian curvature. We have a circle and a line for cylindrical tangency while if k1=k2 we have instead of 1-spheres a 4-blade 2-sphere for umbilical p or a 4-blade 2-plane when k1=k2=0 at flat p.
    More generally the geometric interpretation of the curvature of an M-curve is M-1 orthogonal 1-sphere "osculating circles" (or tangent lines) meeting at p, though with some degenerative "merginging" into higher order spheres and planes when curvatures are equal and principle directions undefined. However, we can always sum the blades to obtain a meaningful single multivector curvature representation, transparently handling degenerecies if correctly implemented. The principle circles have as radii the reciprocals k1-1, k2-1,... of the principle curvatures and for M³2 the point p is recoverable from the curvature as the meet of the 1=spheres.

Curvature Measures
    Curvature measures evaluate or estimate the average or representative curvature of an M-curve over a small Borel neighbourhood within the M-curve of a point p on the M-curve, such as over a "geodesic disk". The sclar Gaussian and mean curvatures are easily measured as scalar integrals over the neighbourhood but measurement of the full Curvature tensor requires some form of anisotropic curvature measure, typically a 3x3 symmetric matrix for M=2 in N=3, which can be integrated using standard matrix addition, and then diagonalised to recover the geometric content.
    Since we know how to add circles in the conformally embeeded Geometric Algebra, a natural anisotropic curvature measure is the ortho 1-spheres 3-vector representation. Summing multiple 3-blades gives a Â4,1 3-vector dual to a 2-vector which can be decomposed into two commuting "principle measure" circles and lines which may meet as a seperating 1-sphere rather than at p, providing a meaure of the spread of our sampling around p 3-blades

Curvature Mesaures of Polyhedral 2-curves
    For a planar polyedral "mesh" approximation to a 2-curve, the Gaussian curvature at vertex p is the angle defect at that vertex, ie. 2p minus the summed angles between consequetive edges incident on p. The contibution to the mean curvature along an edge is constant and is the angle subtended by the triangular planar elemts incident on the edge, considered negative for concave incidence. The contribution to the Gaussian curvature along a straight edge is zero.
    The geometry introduced into matrix bilinear curvature measures along a straight edge with the matrix vector square   EE of the unit edge direction E and those of the normalised sum and difference of adjacent facet normal vectors [ see Cohen-Steiner Definition 4].
    Geometrically, we consider each edge incidnt to p to contribute a cylindrical mean curvature b along the edge, considering the edge as tangent to a cylinder of radius ½b-1. Strictly speaking we should integrate S(P) + E along the edge element   to accrue the correctly spread  positional measure meet but we can as an approximation place a single 1-sphere half way along the edge element and weight by the edge length   

Curvature as loop integative limit
    If [ui p,ujp]=0 so that we have a loop  -Dli¬j  -Dli¬i  Dlj¬j  Dli¬i  p = p then  
    cpui pÙujp(uk p) = (DliDlj)-1  j­-Dlj(  i­-Dli(  j­Dlj(  i­Dli(uk p) ))))
which is clearly dependant solely on value of uk p and the geometry of CM . In particular, it is independant of any differential of uk p.
    We thus have the geometric interpretation of cpui pujp)(uk p) as the 1-vector change in uk p resulting from parallel transport of uk p around (the projection into CM of) a tiny planar loop through p in ui pÙujp ,  divided by the "area" of that loop. This must lie in Ip so we have
    cpui pÙujp(uk p) = ål=14 clijkulp     where scalar clijk º ulp¿cpui pÙujp(uk p) .
[ Proof : j­-Dlj i­-Dli j­Dlj i­Dli(uk p) = e-DljÐlj e-DliÐli eDljÐlj eDliÐliuk p = ...[tedious expansion]... = uk p + 2DliDlj(Ðli×Ðlj)uk p + O(Dl3)  .]

Curvature as Coderivative Operator

    The second directional coderivative operator ÐliÐlj is not symmetric in i,j so we have the Rieman Curvature operator
    cpui pÙujp(u)   =   cpui pÙujp×u   =   2(Ðli×Ðlj)u = ( Я(li)wujp - Я(lj)wui p + wui p×wujp) . u
[ Proof :
ÐbÐau = ( Я(b) + wb×)( Я(a) + wa×)u
= ( Я(b) + wb×)( Я(a)u + ½(wau-uwa)) = ( Я(b) Я(a)u + ½( Я(b)wau- Я(b)uwa) +wb×( Я(a)u +   ½(wau-uwa))
= Я(b) Я(a)u + ½( Я(b)wau - Я(b)uwa + wb Я(a)u - Я(a)uwb ) + ½wb×(wau-uwa)
= Я(b) Я(a)u + ½( Я(b)(wau) + wb Я(a)u - Я(a)(uwb) - Я(b)(uwa) ) + ¼( wbwau - wbuwa - wauwb + uwawb )
Þ 2(b×a)u = 2( Я(b)× Ð¯(a) + Я(b)×(wa×) + (wb×)× Ð¯(a) + (wb×)×(wa×))u
= Я(b)(wau)- Я(a)(wbu) + wb Я(a)u -wa Я(b)u + (wb×wa)×u = ( Я(b)wa)u - ( Я(a)wb)u + (wb×wa)×u
= ( Я(b)wa)×u - ( Я(a)wb)×u + (wb×wa)×u

    Given a linear connection, the second-derivative curvature operator  cpaÙb(v) is thus a geometric commutatator product with a particular bivector cpaÙb .

    The more general definition of the curvature operator    is 2(Ða×Ðb) - Ð[a,b] but we are most interested in the case when [a,b]=0 ( ie. Ða×Ðb =0 ) .

    cpaÙb º Я(b)wa - Я(a)wb + wb×wa =( )= ( Я(b)wa) - ( Я(a)wb) + (wb×wa)     is known as the curvature 2-tensor (aka. Riemann-Christoffel tensor). It is a CM-point-dependant Ip-bivector-valued function of Ip-bivectors acting as a "second order correction" for the linear approximator ¦(p0) + ¦Ñp(dp) for ¦ near p0.

Symmetries and Bianci Relations
    It is possible to express Cxeij in terms of the metric gij and their derivatives, revealing further symmetries:
    Clkij º el¿(Cxeij(ek)) = ei¿(Cxelk(ej)) = -ek¿(Cxeij(el))   .
    The last of these reflects the reversability of parallel transport.
    We also have (from the G expression for Cijkl) the first Bianci identity
   ek¿Cxeij + ej¿Cxeki + ei¿Cxejk = 0 .
    which can also be expressed as     Cxeij ¿ ekl = Cxekl ¿ eij .
    It follows that a¿Cx(bÙc)) + b¿Cx(cÙa) + c¿Cx(aÙb) = 0     " 1-vector a,b,c Î Ip.
    As a result, the M 4 element matrix representation of Cx has only M 2(M 2-1)/12 independant elements (20 out of 256 for N=4).

    ÐeiðCxejk + ÐejðCxeki + ÐekðCxeij = 02         is known as the second Bianci identity.

    The codivergence of the Ricci tensor Ñðx¿ cpa º åiei¿Ðei cpa = ½ÐaCx
[ Proof : Ñðx¿ cpb = Ñðx¿(Ña¿cpab)) Þ ... ???  .]   

    The differential operator Ða Я(b) - Ðb Я(a) - 2( Я(a)× Ð¯(b)) is known as the tortion operator.   It is 0 for a symmetric connection when we thus have
    ui pujp = Ðui p Я(ujp) - Ðujp Я(ui p) .
    Suppose now that ui p is a geodesic field and 1-vector bp is Lie dragged along geodesic ui p congruence so that ui pbp = 0 . We must then have
    Ðui p2 Я(bp) = 2(Ðui p×Ðbp) Я(ui p) .
[ Proof :  Ðui p Я(bp) = Ðbp Я(ui p) Þ Ðui p2 Я(bp) = Ðui pÐbp Я(ui p) = 2(Ðui p×Ðbp) Я(ui p) + ÐbpÐui p Я(ui p)   = 2(Ðui p×Ðbp) Я(ui p) since ui p geodesic Þ Ðui p Я(ui p) = 0  .]
    Thus we have a second geometric interpretation of the curvature tensor as the second derivative of a vector dragged along a geodesic confluence, informally: a measure of geodesic "splay".

[Under Construction]

Sierpinski (<1K) References/Source Material

    David Hestenes "New Foundations For Mathematical Physics" Websource

    Bernard Schutz "Geometrical Methods of Mathematical Physics" Cambridge University Press 1980 [Amazon US UK]
[ Traditional presentation of manifold derivatives]

    David Hestenes, Garret Sobczyk "Clifford Algebra to Geometric Calculus" D. Reidel Publishing 1984,1992 [Amazon US UK]
Since this document draws heavily from and frequently cites this work we clarify the notational differences between this document and the seminal work.
    Their P(a) is our ¯Ip(a) º ¯(a) ("projector").
    Their Pb(a) is our ¯Ñ(a,b) ("differential of projector").
    Their Sa is our [ÑÙ¯](a) ("curl").
    Their Ñ is our Ñ  ("coderivative").
    Their is predominantly our Ñ ("tangential derivative") but also our Ñ ("derivative") early in the work.
    Their da is our Ðaß º ¯Ð¯aß ("extensor coderivative").
    Their (a.) is our Ða=(a¿Ñ) ("directed derivative").
    Their ei is our hip  (potentially nonorthonormal "tangent frame").
    Their gi is our ei (orthonormal "fiducial (basis) frame").

    David Cohen-Striener, Jean-Marie Morvan Restricted Delaunay Triangulations and Normal Cycles 2003.

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Copyright (c) Ian C G Bell 1998, 2014
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