M-field I_p can be expressed in U_N via ^NC_M scalar fields i_ij..mp = e^ij...m¿I_p though the normalisation condition renders one such field redundnat apart from sign. In the mapspace ¦^Ñ-1(I_p) is everywhere the map pseudoscalar.
    In U_N 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 c_paÙb, as a bivector valued function of bivectors, constarined within and upon I_p requires ^MC₂² = ¼M²(M-1)² scalar fields in theory although the Bianchi symmetries reduce the generality considerably. At a fixed given p Î C_M,  mapping bivectors in I_p to bivectors in I_p induces the obvious C_x(aÙb) = ¦^-Ñ_p(c_{p¦^Ñp(aÙb)}) 2-tensor acting in the mapspace which we can represent with ¼M²(M-1)² scalar fields C^kl_ij º e^kl¿C_x(e_ij)

   
    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 C^k_ijl º e^k¿(C_x(e_ij)×e_l)


     c_paÙb and C_x(aÙb) are alternate representations of the same "thing", although c_p defined over subspace I_p of U_N can more readily be "extended" out of C_M.     We can contract the curvature tensor to obtain the Ricci 1-tensor)
    c_pb º Ñ_a¿c_paÙb = å_i=1⁴ eⁱ¿c_peiÙb = å_i=1⁴ eⁱ¿ (( Ð_¯(b)w_ei) - ( Ð_¯(ei)w_b) + (w_b×w_ei))    , 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 C^k_j º e^k¿ C_x(e_i)   = å_j=1^M C^kj_ij or with C_ij º å_k=1^M C^k_ijk which itself contracts to the scalar curvature R = å_i=1⁴ Cⁱ_i .
    The mapspace Einstein tensor is given by M² scalar fields   G^k_j º C^k_j - ½R .

Streamline Coordinates


Introduction
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 v_p defined over M-curve C_M such that v_p Î I_p . It follows from a mathematical result concerning the unique solvability of ordinary differential equations that, provided v_p is sufficiently smooth and is nowhere 0-valued, for every p₀ÎC_M there is a unique "tangent matcher" path in U^N for v_p :
    C_{i p} = { p(t) : t Î [t₀,t₁] }  [ where t₀<0<t₁ ] with p(0)=p₀ ; p(t) Î C_M and  (¶/¶t)p(t) = v_p(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 v_p. We will use both terms interchangeably here. Streamlines cannot intercept eachother (other than at points with v_p=0 which we have disallowed) . Since every pÎC_M has such a curve, the set of all integral curves for each v_p (known as a congruence for v_p) "fills" and effectively defines C_M.
    It is important to note that the 1-curve streamlines of field v_p are independant of the magnitude of v_p, (given v_p² 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 u_{i p} defined over M-curve C_M satisfying
    u₁(p)Ùu₂(p)Ù..u_M(p) = a_pI_p     " p Î C_M where scalar field a_p > 0 " p .
    An obvious example is u_{i p}=¦^Ñ_x(e_i)= h_ip  where C_M=¦(M_ap) is a point embedding specification for C_M.

    Given a general multivector-valued function F(p) defined over C_M , u_{i p} generates at each pÎC_M a specific "streamlined" derivative of F which is the tangential derivative within the 1-curve u_{i p} streamline. We will write this as
    d_li(F(p)) º  ¶F(p_i(l_i))/¶l_i . We will sometime ommit the brackets and write d_liF(p).
    If we so differentiate the identity map F(p)=1(p)=p we recover u_{i p} and, indeed, one can think of 1-field u_{i p} as being the streamlined derivative operator field d_lip . We will accordingly use   the symbol Ð_{¯(ui p)} interchangeably with d_li .

    We introduce the notation  _Dli¬ⁱ p to indicate the point p_i(Dl_i) Î C_M reached by moving from p by arclength Dl_i along the l_i (ie. the u_{i p}) streamline through p.
     _Dli¬ⁱ p º ò_{Ci p [0,Dli]} 1 dp = ò₀^Dl_i p_i'(l_i) dl_i
    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 Dl_i within the neighbourhood of ¦ applicability. For small Dl_i=e we can apply Taylor's Theorem to the streamline l_i-parameterisation and obtain F( _e¬ⁱ p) = e^ed_liF(p) and in particular [  Set F(p)=p ] we have  _e¬ⁱ p = e^ed_li(p) which we can write as
     _e¬ⁱ  = e^ed_li .

    We might be tempted to try to parameterise C_M using y:M_ap®C_M defined by
    y(Dl₁, Dl₂, ..Dl_m, q₀) º  _Dlm¬^M  _Dlm-1¬^M-1 ... _Dl2¬²  _Dl1¬¹ q₀ =_( )=  _Dlm¬^M ( _Dlm-1¬^M-1 (... _Dl2¬² ( _Dl1¬¹ q₀))..) .
    [   That is, starting from a fixed reference point q₀ÎC_M we follow the v₁(p) stream line for arclength Dl₁ to a new point  _Dl1¬¹ q₀ from which we follow the v₂(p) streamline for arc length Dl₂ to  _Dl2¬²  _Dl1¬¹ p₀ and so forth. After M such steps we arrive a final point q_MÎC_M which we can associate with parameters (Dl₁,..,Dl_m). ]
    Any M scalars (sufficiently small that we stay within the neighbourhood) interpreted in this way define a point in C_M, 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 C_M.
    Only if the d_li (and hence the  _Dli¬ⁱ ) everywhere commute with eachother do the l_i provide a true coordinate system.
[ Proof : ...  .]

    ( _e¬ⁱ  ×  _e¬^j ) is a 1-field defined over C_M returning a U_N 1-vector ( _e¬ⁱ  ×  _e¬^j )(p) = (e^ed_li × e^ed_lj)(p) = ... = e²(d_li×d_lj)(p)   + _O(e³) .
    Taking e®0 we can visualise  (d_li×d_lj)(p) º [ u_{i p},u_jp ](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
    [u_{i p},u_jp](v) º  d_li(d_lj[v]) -  d_lj(d_li[v]) = 2(d_li×d_lj)v
which we can write as [u_{i p},u_jp] = d_liu_jp - d_lju_{i 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 u_{i p} generates a congruence for C_M so let us fix on u_{i p} as a "preferred" congruence with associated dervivative field d_li . Consider two points p₀ and p₀ in C_M with p₀= _Dli¬ⁱ p₀ for small Dl_i>0. Clearly p₀= _-Dli¬ⁱ p₀ .
    For j¹i, the specific u_jp streamline through p₀ (i¹1) at p with associated directed derivative d_lj generates a distinct path p_j^*(l_j) =  _-Dli¬ⁱ p_j(l_j) passing through point p₀ =  _-Dli¬ⁱ p₀ known as the Lie drag of u_jp . [  Here ^* indicates "new" or "modifed" rather than a multivector dual ] . This path need not be the streamline at p₀ of any of our u_{k p} 1-fields and, in particular, it need not be the l_j streamline there .
    It is  however, a streamline having at p₀ a tangent vector written (ⁱÝ_-Dli(u_jp))(p₀) fully within I_p0 and an associated streamlined differential operator d_lj^* = (ⁱÝ_-Dli(d_lj))(p₀)  .
    Though d_lj^* is not one of the d_lk, it is not general. Because of its construction it commutes with d_li (ie. d_lj^*(d_li(F))) = d_li(d_lj^*(F))) . [ Proof : ....  .]
    We refer to     (ⁱÝ_-Dli(u_jp))(p₀)     ( or (ⁱÝ_-Dli(d_lj))(p₀) ) as the Lie drag of u_jp ( or d_lj ) from p₀ to p₀=  _-Dli¬ⁱ p₀ .
    We define the Lie drag of a scalar field f(p) defined over C_M by (ⁱÝ_-Dli(f))(p) º f( _-Dli¬ⁱ p) so that (ⁱÝ_e(f))(p) = e^ed_lif .                                              7

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

    Since the Lie drag of u_jp (or d_lj) from p₀ to p₀ can be meaningfully subtracted from u_{i p} (or d_li) at p₀ . The Lie dervivative at p₀ of u_jp with regard to 1-field u_{i p} is an operator defined (by its action on a general 1-field F) as
    ( £_dli(d_lj))F(p) º Lim_{Dli ® 0} [ ( (ⁱÝ_-Dli(d_lj))(p₀)(F(p)) - d_lj(F(p)) )  / Dl_i ] ï_p0 = Lim_{Dli ® 0} [2(d_li×d_lj)F  ï_p0 ]
[ Proof : d_lj^*F ï_p0 = d_lj^*F ï_p0 - Dl_i(d_li(d_lj^*(F))))ï_p0 + _O(Dl_i²) = d_lj^*F ï_p0 + Dl_i(d_li(d_lj(F))ï_p0 + _O(Dl_i²) - Dl_i(d_lj^*(d_li(F))ï_p0 + _O(Dl_i²)
    Hence Lim_{Dli ® 0} [ ( (ⁱÝ_-Dli(d_lj))(p₀) - d_lj )F  / Dl_i ] ï_p0 = Lim_{Dli ® 0} [((d_lj^* - d_lj )F  ï_p0 ) / Dl_i ] = Lim_{Dli ® 0} [(d_li(d_lj(F)) - d_lj^*(d_li(F)) ) ï_p0 ] which provides the result assuming d_lj^* can be safely replaced in the limit by d_lj  .]
    Omitting the brackets , we thus have the 1-vector operator identity
    £_{ui p}u_jp = £_dlid_lj = 2 d_li×d_lj = [ u_{i p},u_jp ] .

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

Covectors

    Any "inner product"  ¿ is a (0;2)-tensor (a potentially point-dependant bilinear scalar-valued function of two 1-vectors in V^M) . For any 1-field ( (1;0)-tensor) v_p , ¿ induces a scalar-valued directional function ( a (0;1)-tensor)     v_p(a) º a¿v_p     known as a covector field. We can recover v_p at a given p from its covector as v_p = Ñ_av_p(a) . A vector and its covector are essentially alternate representations of the same "thing". If we "feed" v_p "to" its covector we get "squared magnitude" 0-tensor v_p(v_p) = v_p¿v_p
    For ¿ in Â_N , if 1-vector v_p is regarded as a 1×N matrix "column vector" , then its covector is the N×1 matrix "row vector" (or transpose) v_p^T . For ¿ in Â_p,q , we take the transpose of the conjugate v_p^† .
    With regard to ¿, we will here define the 1-covector of v_p = å_i=1^N  vⁱe_i as the 1-vector v_pc º  å_i=1^N vⁱe_i so that v_pc ¿₊ a =    v_p¿a .
    In a Euclidean space, v_pc = v_p .
    (1;0)-tensor v_p has "contravariant" coordinate representation å_ivⁱe_i where vⁱ º eⁱ¿v_p .
    (0;1)-tensor v_p has "covariant" coordinate representation å_iv_ie ⁱ where v_i º v_p(e_i) = e_i¿v_p and e ⁱ is the 1-covector of e_i 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 qÎC_M a function ­_q^p C(a) Î I_p giving the tangent vector at pÎC_M "parallel" to aÎI_q at q ÎC_M ; 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. ⁱ­^e_p(a) ¿ ⁱ­^e_p(b) = a¿b and we can acheive this by insisting that u_{i p}¿w_{ui p a p}(e) = _O(e²) .
    ⁱ­^e_p(a) º ae^{w_{ui p a p}(e)} = a + a¿w_{ui p a p}(e) + _O(e²)
where for given scalar e, w_{ui p a p}(e) is a pure U_N-bivector 2-field defined at every pÎC_M for all aÎI_p .
    Parallel displacement along the i^th streamline thus preserves orthonormality and corresponds to a Lorentz rotation having U_N rotor R_{p 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 C_M.

Linear Connection

    g_p(u_jp,d_li,e) º u_jp - ⁱ­^e_p(u_jp) = -u_jp¿w_{ui p ujp p}(e)  + _O(e²)
      is a 1-vector lying in general neither in I_p 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 Lim_{e ® 0}[ _e¬ⁱ u_jp - ⁱ­^-e _e¬ⁱ p( _e¬ⁱ u_jp)] = -g_p(u_jp,u_{i 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 Â_1,3) fiducial frame is unfortunate. ]
    Taking g_p(u_jp,u_{i p}) º Lim_{e ® 0}[e^-1g_{j p}(u_{i p},e) ] we obtain the connection, the 1-vector component of d_li u_jp ï_p that exists outside I_p.
    g_p(u_jp,u_{i p}) = -u_jp¿w_{ui p ujp} = w_{ui p ujp} ×u_jp .
    g_p(u_jp,u_{i p}) is orthogonal to both u_{i p} and u_jp at p .

    A linear connection (aka. affine connection) derives from an assumed bilinearity of g_p which can then can be fully defined by M² 1-fields   g_p( h_jp , h_ip ) which then effectively define ­_q^p C(a). Indeed, some authors define ⁱ­^Dl_i as the consequence of a given linear connection.
    We say the connection is symmetric if g_p(u_{i p},u_jp) = g_p(u_jp,u_{i p}) .

    Suppose now that the u_{i p} are everywhere mutually orthogonal so that {u_{i p}} provides an orthonormal frame for I_p with inverse frame {uⁱ_p}. Because w_{ui p ujp} is a 2-blade perpendicular to I_p, we have w_{ui p ujp} ×u_{i p} = 0     i¹j . Thus we can define M pure bivectors w_{ui p p} º å_j=1^M w_{ui p ujp} which satisfy g_p(u_jp,u_{i p}) =  w_{ui p p}×u_jp
    The connection is symmetric if u_jp¿w_{ui p p} = u_{i p}¿w_{ujp p}     " 1£i,j£M .
    A linear connection arises if w_{ui p a} is linear in a.
[ Proof : Since the u_{i p} are orthogonal g_p((å_j=1^M b^ju_jp),(å_i=1^M aⁱu_{i p})) = (å_j=1^M b^ju_jp)¿w_{ui p åi 1M=} aⁱu_{i p})}
= (å_j=1^M b^ju_jp)¿(å_i=1^M aⁱw_{ui p ui p} ) = å_i=1^M å_j=1^M aⁱb^j(u_jp¿w_{ui p ui p} ) = å_i=1^M å_j=1^M aⁱb^jg_p(u_jp,u_{i p})  .]

Within the mapspace
    Parallel transport in the mapspace ⁱ­^e_x(b) º ¦^Ñ_x+eei^-1( ⁱ­^e_¦(x)(¦^Ñ_p(b)) ) preserves ¿ instead of ¿.
[ Proof : ⁱ­^e_p(a) ¿_x+eei ⁱ­^e_p(b) = ⁱ­^e_p(¦^Ñ_x(a))¿ⁱ­^e_p(¦^Ñ_x(b)) = ¦^Ñ_p(a)¿¦^Ñ_x(b) = a¿_xb  .]

    Let b be a map direction at xÎM_ap with corresponding C_M direction a=¦^Ñ_x(b) at p=¦(x). It may not make any geometrical "sense" to subtract a map 1-vector "at" x+ee_i from one "at" x but we can still do it!
    G_x(b,e_i,e) º b - ⁱ­^e_x(b) = b - ¦^Ñ_x+eei^-1(ⁱ­^e_p(¦^Ñ_x(b))) = b - ¦^Ñ_x+eei^-1(¦^Ñ_p(b) - g_p(¦^Ñ_p(b),d_li,e))
is, for given e_i and scalar e,  a map-1-vector-valued function of map-1-vector b, linear in b if the C_M connection g_p is linear. We then have G_x(e_i,e_j,e) = å_k=1^M  G^k_ijx(e)e_k . Letting   G_x(b,e_i) =   Lim_{e ® 0}(e^-1G_x(b,e_i,e)) we achieve an affine connection for the mapspace, symmetric if g is.
    G_x(a,b) = ¦^-Ñ_x(¦^Ñ2_x(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¬i p) + DliÐlivj( Dli¬i p) + Dli2Ð (ui p)2vj( Dli¬i p) + ...  .]

    Ð_a can be regarded as a differential over and "within" C_M, although its component parts Ð_a and w_a× cannot.

    Ð_a generates a 1-vector codel-operator
    Ñ_p º å_k=1^M u^k_pÐ_{uk p} = Ñ_[CM] + å_k=1^M u^k_p(w_{uk p}×) .

Within the mapspace
    Ð_a induces a differential operator within M-D parameter space M_apÌV_M
    Ð_a^ð(B(x)) = ¦^-Ñ_p(Ð_¦^Ñp(a)(b(p)))     where b(p) º B(¦^-1(p)) .
    Ð_a^ð(v_p) = Ð_a(v_p) + G(a,v_p)     where G is linear as a consequence of the linearity of ¦^-Ñ_p .
    In particular,
    G_p(e_j,e_i) º Ð_ei^ðe_j = å_k=1^M G^k_jie_k     where point-dependant scalar G^k_ji is known as a Christoffel symbol.

Geodesics
    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 C_M. The equivalence of the two defintions is intuitively reasonable and the proof (ommitted here) arises from fairly straightforward calculus

Curvature
[Under Construction]

1-Curvaturem
    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 Lim_{d ® 0}[ d^-1 q_Ð(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/¶x¹¶x²) 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 h_1p  = ¶p/¶l₁ , h_2p  = ¶p/¶l₂ then
    ¶²p/¶l_i¶l_j º ¶ h_ip /¶l_j = å_k=1² G^k_ij h_kp  + b_ij h_3p      i,j,k Î {1,2}
    ¶ h_3p /¶l_i = - å_{k l=1}² g^klb_li h_kp      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 Â_4,1 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   EºE 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 [u_{i p},u_jp]=0 so that we have a loop  _-Dli¬^j  _-Dli¬ⁱ  _Dlj¬^j  _Dli¬ⁱ  p = p then  
    c_{pui pÙujp}(u_{k p}) = (Dl_iDl_j)^{-1  j}­^-Dl_j(  ⁱ­^-Dl_i(  ^j­^Dl_j(  ⁱ­^Dl_i(u_{k p}) ))))
which is clearly dependant solely on value of u_{k p} and the geometry of C_M . In particular, it is independant of any differential of u_{k p}.
    We thus have the geometric interpretation of c_{pui p}u_jp)(u_{k p}) as the 1-vector change in u_{k p} resulting from parallel transport of u_{k p} around (the projection into C_M of) a tiny planar loop through p in u_{i p}Ùu_jp ,  divided by the "area" of that loop. This must lie in I_p so we have
    c_{pui pÙujp}(u_{k p}) = å_l=1⁴ c^l_ijku_lp     where scalar c^l_ijk º u^l_p¿c_{pui pÙujp}(u_{k p}) .
[ Proof : ^j­^{-Dl_j i}­^{-Dl_i j}­^{Dl_j i}­^Dl_i(u_{k p}) = e^-Dl_jÐ_lj e^-Dl_iÐ_li e^Dl_jÐ_lj e^Dl_iÐ_liu_{k p} = ...[tedious expansion]... = u_{k p} + 2Dl_iDl_j(Ð_li×Ð_lj)u_{k p} + _O(Dl³)  .]

Curvature as Coderivative Operator

    The second directional coderivative operator Ð_liÐ_lj is not symmetric in i,j so we have the Rieman Curvature operator
    c_{pui pÙujp}(u)   =   c_{pui pÙujp}×u   =   2(Ð_li×Ð_lj)u = ( Ð_¯(li)w_ujp - Ð_¯(lj)w_{ui p} + w_{ui p}×w_ujp) . 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  c_paÙb(v) is thus a geometric commutatator product with a particular bivector c_paÙb .

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

    c_paÙb º Ð_¯(b)w_a - Ð_¯(a)w_b + w_b×w_a =_( )= ( Ð_¯(b)w_a) - ( Ð_¯(a)w_b) + (w_b×w_a)     is known as the curvature 2-tensor (aka. Riemann-Christoffel tensor). It is a C_M-point-dependant I_p-bivector-valued function of I_p-bivectors acting as a "second order correction" for the linear approximator ¦(p₀) + ¦^Ñ_p(dp) for ¦ near p₀.

Symmetries and Bianci Relations
    It is possible to express C_xeij in terms of the metric g_ij and their derivatives, revealing further symmetries:
    C^l_kij º e^l¿(C_xeij(e_k)) = eⁱ¿(C_xelk(e_j)) = -e^k¿(C_xeij(e_l))   .
    The last of these reflects the reversability of parallel transport.
    We also have (from the G expression for Cⁱ_jkl) the first Bianci identity
   e_k¿C_xeij + e_j¿C_xeki + e_i¿C_xejk = 0 .
    which can also be expressed as     C_xeij ¿ e_kl = C_xekl ¿ e_ij .
    It follows that a¿C_x(bÙc)) + b¿C_x(cÙa) + c¿C_x(aÙb) = 0     " 1-vector a,b,c Î I_p.
    As a result, the M ⁴ element matrix representation of C_x has only M ²(M ²-1)/12 independant elements (20 out of 256 for N=4).

    Ð_ei^ðC_xejk + Ð_ej^ðC_xeki + Ð_ek^ðC_xeij = 0₂         is known as the second Bianci identity.

    The codivergence of the Ricci tensor Ñ^ð_x¿ c_pa º å_ieⁱ¿Ð_ei c_pa = ½Ð_aC_x
[ Proof : Ñ^ð_x¿ c_pb = Ñ^ð_x¿(Ñ_a¿c_pab)) Þ ... ???  .]   

Tortion
    The differential operator Ð_a Ð_¯(b) - Ð_b Ð_¯(a) - 2( Ð_¯(a)× Ð_¯(b)) is known as the tortion operator.   It is 0 for a symmetric connection when we thus have
    £_{ui p}u_jp = Ð_{ui p} Ð_¯(ujp) - Ð_ujp Ð_{¯(ui p)} .
    Suppose now that u_{i p} is a geodesic field and 1-vector b_p is Lie dragged along geodesic u_{i p} congruence so that £_{ui p}b_p = 0 . We must then have
    Ð_{ui p}² Ð_¯(bp) = 2(Ð_{ui p}×Ð_bp) Ð_{¯(ui p)} .
[ Proof :  Ð_{ui p} Ð_¯(bp) = Ð_bp Ð_{¯(ui p)} Þ Ð_{ui p}² Ð_¯(bp) = Ð_{ui p}Ð_bp Ð_{¯(ui p)} = 2(Ð_{ui p}×Ð_bp) Ð_{¯(ui p)} + Ð_bpÐ_{ui p} Ð_{¯(ui p)}   = 2(Ð_{ui p}×Ð_bp) Ð_{¯(ui p)} since u_{i p} geodesic Þ Ð_{ui p} Ð_{¯(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]

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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 P_b(a) is our ¯^Ñ(a,b) ("differential of projector").
    Their S_a 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 d_a is our Ð_aß º ¯Ð_¯aß ("extensor coderivative").
    Their (a.¶) is our Ð_a=(a¿Ñ) ("directed derivative").
    Their e_i is our h_ip  (potentially nonorthonormal "tangent frame").
    Their g_i is our e_i (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
Web Source: www.iancgbell.clara.net/maths
Latest Edit: 04 Oct 2014.