Manifold Restricted Tensors

Further differentials and derivatives
To the extent that attention is focused on intrinsic properties of the manifold, Ñ is fragmented and various brands of derivative emerge. If one "renounces" the extrinsic orientation M-blade pseudoscalar, one is left with the intrinsic affine connection and associated Riemann and Ricci curvature tensors which tend to be mathematically and notationally fearsome. Nonetheless, we shall proceed some way along this route. The casual reader should skip to the next chapter.

Operator Notations
If f1(a) is an operator taking one parameter then f1×(a,b) º ½(f1(a)×f1(b))
If f2(a,b) is an operator taking two parameters then f2×(a,b) º ½(f2(b,a)-f2(a,b))
f1<k>(a) º f1(a<k>)

We define f1¨ by f1¨ (a) ¨ a = f1(a) where ¨ is a multivector product.
Thus, for example, if tensor fp(a) is scalar valued then ¦¿(a) is the 1-tensor f(a) = (f(a))a (a-2) satisfying f(a)¿a = f(a) .

There is a strong temptation to abbreviate ¦(a) as ¦a , "ommiting the brackets", but this is dangerous because ¦a more properly denotes the composite operator (¦(a))(b) = ¦(ab) .
Confusion can arise in particular in the case of an operator like Ñp which we also consider to be "like" a 1-vector geometrically. f(Ñ) and fÑ are then fundamentally different constructs.
We will insert a composition product symbol ° between two operators only when we wish to emphasise an "assumed" compositional product. The composition product symbol will normally be ommitted for brevity. Thus
f°g(a) º f(g(a)) º (fg)(a)   º fg(a)

How are we to interpret f1Ñg1 º (f1Ñ)g1 ? The "differentialiser" Ñ converts f1 from an operator or tensor taking one nonprimary argument to one taking two, it introduces a second non-primary 1-vector parameter. In nonabbreviated expressions we will add this parameter at the rightmost end of the parameter list with
f1Ñ(a,b) = (Ðbf1)(a) = Ðb(f1(a)) - f1(Ðba)  . We will interpret the g1 in the "composite product" f1Ñg1 as applying to the "newest" rightmost nonprimary parameter so that f1Ñg1(a,b) º f1Ñ(a,g1(b)) .
From a programmers' perspective, we can think of "parsing" our composite operator expressions from left to right pushing introduced parameters onto a stack. When a "compositional irregularity" such as f1Ñg1 is encoutered it is the most "recent" parameter to which they apply in accordance wth a stanard "last in first out" stack. We will accordingly refer to f1Ñg1(a,b) º f1Ñ(a,g1(b)) as the LIFO convention.

Tangential 1-differential  Ñ

We can define the tangential 1-differential of field Fp by
FÑ[CM]pp(a) º FpÑ(a) º (a*Ñ[CM]p)F(p)   =   (a*¯(Ñp))F(p)   =   (¯(a)*Ñp)F(p)   =   FpÑ(¯(a)) º FpÑ¯(a) which gives us
Ñ   =   Ñ¯     abbreviating FpÑ(a) = FpÑ¯(a) º FpÑ(¯(a)) .
; the a-directed tangential derivative is equivalent to the ¯(a)-directed derivative.

Consider now (ÑFp)Ñ(a) = ¯Ñ(Ñb,a)FpÑ(b) + ÑbFpÑ2(b,a)     [ HS 4-1.18 ]

More generally suppose Fp(a1,a2,...ak) is an extensor tensor of k multivector variables themselves p-dependant. We can take a 1-differential of the secondary differntial Ña1Fp(a1,a2,...ak) to give
(Ña1Fp)Ñ(a1,..,ak,b) = ¯Ñ(Ña1,b)Fp(a1,..,ak) + Ña1FpÑ(a1,..,ak,b) .
For 1-vector a1 = ¯a1 we can write this as
(b*Ñ)ÑÞ   =   ¯Ñ(ÑÞ, b)     + ÑÞ(b*Ñ)     which we will here refer to as the primary-secondary Ñ commutation rule     [ HS 4-1.19b ] .

Analagous to the gradifying substitution rule we have the tangential gradifying substituion rule that applying the operatior (Ñß¿ÑÞ) is equivalent to replacing the first nonprimary argument of a tensor with ÑÜ .
[ Proof :  (Ña¿Ñß)Fp(a,b,..) = åi=1M ei2 Ðaei(ÐpeiFp)(a,b,...) = åi=1M (ÐpeiFp)(ei2ei,b,...) = åi=1M (ÐpeiFp)(ei,b,...)
= åi=1M FpÑ(Ðpeiei,b,...) = FpÑ(Ñ,b,...)  .]

Hypercurve
For M=N-1 we have Ñ = ¯(Ñ) = Ñ - np(np¿Ñ) = Ñ - npÐnp .
Since np¿Ðanp = 0 " a , Ñpnp lies within iN-1p and ^[Ñpnp] = np-1ÙÐnpnp  .
More generally, let np be a unit 1-field. Ñp = np2Ñp = np(np¿Ñp) + np(npÙÑp) corresponding to tangential and orthogonal components of Ñp .

Directed Coderivative      Ð()

With regard to a multivector field Fp defined over CM (but not necessarily confined within Ip) we can most simply define the a-directed coderivative for aÎUN as the projection of the a-directed tangential derivative, ie. the projection of the ¯Ip(a)-directed dirivative
Ð = ¯ Ð¯() º ¯° Ð¯()     abbreviating ÐaFp º ¯IpÐ¯IpaFp

The symbol Ð can thus be thought of as an abbreviation
Ð   =   ¯ Ð¯()   =   Ð¯()¯ - ¯Ñ     abbreviating Ðb(ap) = ¯( Ð¯(b)ap) = Ð¯(b)(¯(ap)) - ¯Ñ(ap,b)     [ HS 5-1.1 ] . The underscore serves to remind us of a p and Ip (ie. CM) dependance.
[ Proof :   Ð¯(d)(¯(ap)) = ( Ð¯(d)¯)(ap) + ¯( Ð¯(ap)) Þ Ðd(ap) º ¯( Ð¯(d)(ap)) = Ð¯(d)(¯(ap)) - ( Ð¯(d)¯)(ap) = Ð¯(d)(¯(ap)) - ¯Ñ(ap)  .]

The directed coderivative of an extended field is defined by
(ÐdFp)(a1,a2,...) º Ðd(Fp(a1,a2,..)) - Fp(Ðda1,a2,...)- Fp(a1,Ðda2,...) = ¯( Ð¯(d)(Fp(a1,a2,..)) ) - Fp(¯( Ð¯(d)a1),a2,...)- Fp(a1,¯( Ð¯(d)a2),...)
Note that this differs from [ HS 4-3.3 ] which inserts subtracts - Fp(Ðd¯(a1),a2,...)- Fp(a1,¯( Ð¯(d)a2),...)

Coderivative     Ñ
The covariant derivative Ða within an M-curve can be approached in a number of creative ways. One is to simply write down all the properties we would like a derivative to have ( such as ÑpFp Î Ip ; Ðpa1p + Ðpa2p = Ðp(a1p+a2p) ; and so forth ) and then intone "as defined so mote it be" three times at midnight. Another is based on projections [see General Relativity ].    Many appeal to a notion of "parallel transport" which is defined (or not) in a variety of ways.
We define the undirected coderivative 1-vector operator as the projected tangential derivative
ÑFp º Ñ[Ip]pFp º ¯Ip(Ñ[Ip]pFp)     which we can express operationally as Ñ º  ¯Ñ     noting carefully that this denotes operator composition ¯°Ñ º ¯(Ñ(ap)) rather than ¯(Ñ)(ap) = ¯(¯(Ñ))(ap) = ¯(Ñ)(ap) = Ñ(ap) . The differentiating scope of the Ñ is to be thought of as extending rightwards only in the usual manner, and not effecting the ¯ .
We have the operator identity Ða = (a¿Ñp) .
Since ÑFp Î Ip for a scalar field Fp the coderivative operator Ñp is equivalent to the tangential derivative Ñp when acting on scalar fields, and for aÎIp the a-directed coderivative Ða is equivalent to the a-directed derivative Ða when acting on scalar fields.

Ñp(ap) º ¯(Ñp(ap)) = Ñp(¯(apÑ))     by the projected product rule .
This is particularly clear when expressed in coordinate terms with a fortuitous basis as Ñp(¯(apÑ)) = åi=1N ei Ð¯(ei)(¯(apÑ)) = åi=1M ei(¯( Ð¯(ei)ap)) = åi=1M ¯(ei)(¯( Ð¯(ei)ap)) = åi=1M ¯(ei( Ð¯(ei)ap)) = ¯(ei(åi=1M  Ð¯(ei)ap)) = ¯(Ñp(Ap)) º Ñp(Ap) .

For an extended field Fp(a1p,a2p,...akp) with aipÎUN we have a-directed primary coderivative
ÐaFp(a1p,a2p,...akp) º (ÐaFp)(a1p,a2p,...akp)
= Ða(Fp(a1p,a2p,...akp)) - Fp(Ða(¯Ip(a1p)),a2p,...akp) - Fp(aip,Ða(¯Ip(a2p)),,...akp) - Fp(aip,a2p,...,Ða(¯Ip(akp)))
and an ith exterior  coderivative
Ñi¿p Fp(a1p,a2p,...akp) º ¯Ip(Ñi¿p Fp(a1p,a2p,...akp)) , the projection of the ith conveyed divergence.

The cogradifying substitution rule that applying the operator (Ñß¿ÑÞ) = (Ñß¿_csgrad) º ¯°(Ñß¿ÑÞ) is equivalent to replacing the first nonprimary parameter with ÑÜ defined by follows immediately as the projection of tne tangential gradifying substitution rule.

We will see that ÑÙÑÙap = 0     " path-independant ap , so that Ñ2 , while not a scalar operator, does not increase grade .

Hypercurve
For M=N-1 we have Ñnp = Ñnp .

Projection Differential (1.2)-tensor     ¯Ñ

¯Ñ(ap,d) º ¯IpÑp(ap,d) º Ðd(¯Ip(ap)) - ¯Ip(Ðdap)     (informally) the rate of change of ¯Ip(a) in direction d , is of of less interest than ¯Ñ,  the primary tangential 1-differential of ¯Ip (ie. the second tangential 1-differential of the identity function) ,
¯Ñp(ap,d)   =   ¯Ñp(ap,¯d)   =   1Ñ2(ap,d)   =   ( Ð¯(d)¯Ip)(ap)     =   Ð¯(d)(¯(ap)) -  ¯( Ð¯(d)ap)   =   2( Ð¯(d)×¯)(ap)  .
The symmetry ¯Ñ(a,b) =¯Ñ(b,a) for a,b Î Ip follows from the symmetry of 1Ñ2 in UN.
Tangentially differentiating the outtermorphism result ¯Ip(aÙb)=¯Ip(a) Ù ¯Ip(b) yields
¯Ñ(aÙb,d) =  ¯Ñ(a,d) Ù ¯(b) +  ¯(a) Ù ¯Ñ(b,d)     [ HS 4-2.6 ] and hence
¯Ñ(a1Ùa2Ù....ak,d) = åi=1k (-1)i+1 ¯Ñ(ai,d)Ù¯Ip(a1)Ù... ¯Ip(ai-1)Ù ¯Ip(ai+1)Ù... ¯Ip(ak) . [ HS 4-2.31 ]

¯¯Ñ   =   ¯Ñ^       ;     ^¯Ñ¯   =   ¯Ñ¯       [ HS 4-2.11 ] and so
¯=¯Ñ   =   ^=¯Ñ   =   0     abbreviating ¯(¯Ñ(¯(a),b)) = ^(¯Ñ(^(a),b)) = 0     " a,b.
[ Proof :  Tangentially differentiating  ¯2(a)  =¯(a) gives ¯Ñ(a,d) = ¯Ñ(¯(a),d) + ¯(¯Ñ(a,d)) Þ ¯¯Ñ(a,d) = ¯Ñ((1-¯)(a),d) = ¯Ñ(^(a),d) .
Thus ¯=¯Ñ = ¯Ñ ^¯  = ¯Ñ 0 = 0  .]

Trivially therefore, ¯¯Ñ(Ip,d) = 0 and hence ¯Ñ(Ip,d)¿Ip = 0     " d .
ÑÞ¿(¯Ñ¯) = 0 abbreviating Ñb¿¯Ñ(¯a,b) = 0 " a,b        [ HS 4-2.17a ] .
[ Proof :  ÑÞ¿¯Ñ(¯a,b)  (¯(ÑÞ))¿¯Ñb) =  ÑÞ¿(¯¯Ñb)) = ÑÞ¿0 .  .]

ÑÞÙ(¯Ñ^) = 0
[ Proof : See [ HS 4-2.17b ]  .]

¯Ñ enables us to express the tangential differential of the undirected derivative in terms of the first and second tangential differential as
(ÑFp)Ñ(a) = ¯Ñ(Ña,b)FpÑ(a)) + ÑaFpÑ2(a,b)     provided aÎIp .
[ Proof : aÎIp Þ FpÑ(a) = (a¿Ñ)Fp = (¯(a)¿Ñ)Fp = (¯(a)¿Ñ)Fp = FpÑ(¯(a))  .]

Hypercurve
For M=N-1 we have ¯Ñ(ap,b) = (Ðb¯)(ap)   = -np2((ap¿npÑ(b))np + (ap¿np)npÑ(b)) .
Thus ¯Ñ(¯(ap),b) = -np2 (¯(ap)¿npÑ(b))np = -np2 (ap¿npÑ(b))np ;
Thus ¯Ñ(np,b) = -npÑ(b) .     , the normalisation condition on np providing np¿npÑ(b)=0 .

Projection Second Differential (1;3)-tensor     ¯Ñ2
We can take the second tangential differential of ¯, ie. the third tangential differntial of 1 obtaining
¯Ñ2(a,b,c) º Ð¯(c)(¯Ñ(a,b)) - ¯Ñ( Ð¯(c)a,b)) - ¯Ñ(a, Ð¯(c)b)) .

Squared Projection Differential (1;3)-tensor     (¯Ñ)2
(¯Ñ)2(a,b,c) º ¯Ñ(¯Ñ(a,b),c)

(¯Ñ)2×¯ = Ð ×¯     abbreviating (¯Ñ)2(¯Fp,b,a) - (¯Ñ)2(¯Fp,a,b) = Ða(Ðb(¯Fp))-Ðb(Ða(¯Fp))
[ Proof :  Recalling ¯=¯Ñ=0 we have:
ÐaÐb(¯Fp) = Ða( Ð¯(b)(¯Fp) - ¯Ñ(¯Fp,b) ) = ¯( Ð¯(a)( Ð¯(b)(¯Fp) - ¯Ñ(¯Fp,b) )) = ¯=( Ð¯(a) Ð¯(b))(Fp) - ¯( Ð¯(a)(¯Ñ(¯Fp,b)) )
= ¯=( Ð¯(a) Ð¯(b))(Fp) - ¯[ ( Ð¯(a)¯Ñ)(¯Fp,b) +  ¯Ñ( Ð¯(a)(¯Fp),b) + ¯Ñ(¯Fp, Ð¯(a)b) ]
= ¯=( Ð¯(a) Ð¯(b))(Fp) - ¯¯Ñ2(¯Fp,b,a) - ¯¯Ñ( Ð¯(a)(¯Fp),b)
= ¯=( Ð¯(a) Ð¯(b))(Fp) - ¯=(¯Ñ2)(Fp,b,a) - ¯¯Ñ( ( Ð¯(a)¯)Fp)+¯( Ð¯(a)Fp),b)
= ¯=( Ð¯(a) Ð¯(b))(Fp) - ¯=(¯Ñ2)(Fp,b,a) - ¯(¯Ñ)2(Fp,a,b)
Þ (Ða×Ðb)Fp = (¯Ñ)2×(Fp,a,b) by symmetry of ¯Ñ2 and integrability condition Ð¯(a)× Ð¯(b)=0  .]

Shape (<0.2>;1)-multitensor     [Ñ¯]
We define the shape [Ñ¯] of CM to be the undirected tangential 1-derivative of the projector , an abbreviation of
[Ñ¯](ap) º Ñß[CM]p¯Ip(ap) º Ñß ¯Ip(ap) = (Ñb Ðp↓(b) ¯Ip)(ap) = Ñb ¯Ñ(ap,b) = Ñb ¯Ñ(ap,b) = Ñb ¯Ñ(ap,b) .

[Ñ¯](ap) = 0 so [Ñ¯] annihilates scalars.

[Ñ¯](ap) decomposes as [Ñ¯](^Ip(ap)) + [Ñ¯](¯Ip(ap)) and, joy of joys, this corresponds precisely to its grade decomposition into scalar and bivector parts. We can express this immense good fortune operationally with regard to general operands as
[Ñ¯] ^   =   ¯ [Ñ¯]   =   [Ñ¿¯]     [ HS 4-2.35b ] which stems from [ÑÙ¯]^ = [Ñ¿¯]¯ = 0 holding generally [ HS 4-2.37 ] ; and
[Ñ¯]¯   =   ^[Ñ¯]    =   [ÑÙ¯]       [ HS 4-2.35a ] since ^[Ñ¯] = [Ñ¯]-¯[Ñ¯] = [Ñ¯]-[Ñ¯]^ = [Ñ¯]¯ .

For general operands we have   [Ñ¯](a)<k+1> = [Ñ¯](¯Ip(a<k>))    ;     [Ñ¯](a)<k-1> = ¯Ip([Ñ¯](a<k>))     and so
Ñ¯ = Ñ¯ + [Ñ¯]¯ = Ñ¯ + [ÑÙ¯]     [ HS 4-3.7a ] with Ñ¿¯ = Ñ¿¯ . Hence
^Ñ¯ = [Ñ¯]¯ = [ÑÙ¯]¯
[ Proof :  Ñ(¯(ap)) = Ñ(¯(ap)) + ^Ñ(¯(ap)) = Ñ¯(ap) + ^( [Ñ¯](ap) + Ñ¯(apÑ) ) = Ñ¯(ap) + ^([Ñ¯](ap) + Ñ(ap))
= Ñ¯(ap) + ^[Ñ¯](ap) = Ñ¯(ap) + [Ñ¯]¯(ap) = (Ñ+[Ñ¯])¯(ap) .  .]

With tangential operands ¯(ap)=ap understood we can write this as
Ñ = Ñ + [Ñ¯] = Ñ + [ÑÙ¯]       [ HS 4-3.6a ] and   ^Ñ = [Ñ¯] = [ÑÙ¯] .

[ÑÙ¯](a) º ([Ñ¯](a))<2> = [Ñ¯]¯(a) is known as the curl (2;1)-tensor .

Scalar ¯1 [Ñ¯](a) =[Ñ¯](^Ip(a)) = (Ñp¿¯Ip)(a) is expressible as (Ñb[ÑÙ¯](b))¿a where 1-vector
Ñb[ÑÙ¯](b) = º ÑÞ[ÑÙ¯](b) = ÑÞ¿[ÑÙ¯](b) º ÑÞ¿[ÑÙ¯](b)

[¯(Ñ)]   =   ¯Ñ(Ñ)   =   ÑÞ [ÑÙ¯]   =   ÑÞ¿[ÑÙ¯]   =   [Ñ¿¯]¿     [ HS 4-2.20 ]
is known as the spur 1-field of CM and lies outside Ip .

[Ñ¿¯](a) = [¯(Ñ)].a [ HS 4-2.18 ] so the shape (<0,2>;1)-tensor is "recoverable" from the curl (2;1)-tensor as [Ñ¯](a) = [ÑÙ¯](a) + (Ñb[ÑÙ¯](b)).a .

1-vector ¯Ip(b)¿[Ñ¯](a) = ¯Ip(b)¿[ÑÙ¯](a) is normal to CM , ie. ¯Ip ( ¯Ip(b)¿[Ñ¯](a)) = 0 .

[Ñ¯](a1Ùa2Ù...ak) = åi=1k.  (-1)i+1 [ [Ñ¯](ai) Ù ¯Ip(a1Ù..Ùai-1Ùai+1..Ùak) + ¯Ip([Ñ¯](ai) × (a1Ù..Ùai-1Ùai+1..Ùak) ]     [ HS 2.41c ] provides the extension of [Ñ¯] to general multivectors.

[Ñ¯]  = ÑÞ2 ¯Ñ     abbreviating [Ñ¯](ap) = Ñb¯Ñb) expressing the shape as the secondary tangential derivative of ¯Ñ [ HS 4-2.14 ].
[ Proof :  Ðbd ¯Ñ(ap,b) = Ðbd( Ð¯(b)(¯(ap)) -  ¯( Ð¯(b)ap)) = Lime ® 0 e-1[ Ð¯(b+ed)(¯(ap)) -  ¯( Ð¯(b+ed)ap)) - Ð¯(b)(¯(ap)) -  ¯( Ð¯(b)ap))]
=   Ð¯(d)(¯(ap)) -  ¯( Ð¯(d)ap)) =  ( Ð¯(d)¯)(ap) so we have both Ñß ¯(ap) º (Ñ¯)(ap) = Ñb¯Ñ(ap,b) º ÑÞ2¯Ñ(ap,b)     and Ñ¯ß ¯(ap) º [Ñ¯](ap) = Ñb¯Ñ(ap,b) º ÑÞ2¯Ñ(ap,b)
Since ¯Ñ(a,b)=¯Ñ(b,a) the result follows.  .]

The grade-tangency associations of [Ñ¯] immediately provide
[Ñ¯]^ = [Ñ¿¯] = ÑÞ2¿¯Ñ     ;     [Ñ¯]¯ =  [ÑÙ¯] = ÑÞ2Ù¯Ñ       [ HS 4-2.16 ].
[ Proof :  [Ñ¯](a) º (Ñp¯))(a) = Ñb¯Ñp(a,b) so .... ???  .]

Ñ2 = [Ñ¯](Ñ) + ¯Ñ2
[ Proof :  Ñ2(ap) =Ñ(¯Ñ(ap)) = [Ñ¯](Ñ) + ¯Ñ(Ñ(ap)) = [Ñ¯](Ñ)(ap) + ¯Ñ(Ñ(ap)) = [Ñ¯](Ñ)(ap) + ¯ÑÑ(ap)
= [Ñ¯](Ñ)(ap) + ¯¯ÑÑ(ap) = [Ñ¯](Ñ)(ap) + ¯Ñ2(ap) = [Ñ¯](Ñ)(ap) + ¯(Ñ¿Ñ)ap  .]

Hence ÑÙÑ = [Ñ¯](Ñ) = [ÑÙ¯](Ñ) with regard to path-independant functions. [ HS 4-3.10b ] .

We have the following Shape Properties:

1.   [ÑÙ¯]^ = 0   ie. [ÑÙ¯](^Ip(ap)) = 0  " ap       [ HS 4-2.37a ]
2.   [Ñ¿¯]¯ = 0 . ie. [Ñ¿¯](¯Ip(ap)) º (Ñ¿¯)(¯(ap)) = 0   " ap     [ HS 4-2.37b ]
3. ¯[ÑÙ¯] = 0     when acting on 1-vectors ie. ¯[ÑÙ¯](a)=0  " a . [ HS 4-2.30 ]
[ Proof : ¯(a)¿[ÑÙ¯](b) = ¯Ñ(¯(a),b) [ HS 4-2.26 ] Þ ¯( ¯(a)¿[ÑÙ¯](b) ) = ¯¯Ñ(¯(a),b)   = ¯=¯Ñ(a,b) = 0 .
Þ ??? ¯( a¿¯([ÑÙ¯](b)) )=0 ????  .]

^[Ñ¿¯] = 0     follows immediately from ^¯1=0 with regard to 1-vectors.
4. I [Ñ¯] = I ×[Ñ¯] = I ×[ÑÙ¯]     abbreviating Ip×[ÑÙ¯](a) = Ip×[Ñ¯](a) = -[Ñ¯](a)×Ip = -[ÑÙ¯](a)×Ip = -[ÑÙ¯](a)Ip    (4-4.5)
[ Proof :  3 Þ [ÑÙ¯](a)¿Ip = 0 and since [ÑÙ¯](a)ÙIp=0 the result follows from bivector nature of [ÑÙ¯]  .]
Hence [ÑÙ¯](a) = ([Ñ¯](a)×Ip)Ip-1 = -Ip-1([ÑÙ¯](a))
Also  Ip.[ÑÙ¯] = IpÙ[ÑÙ¯] = 0     for 1-vector arguments     [ HS 4-4.3 ] which yields
[ÑÙ¯]2 = 0 .
[ Proof :   ¯Ip([ÑÙ¯](a)) º ¯ [ÑÙ¯](a) = 0 Þ [ÑÙ¯](a)=^([ÑÙ¯](a)) Þ [ÑÙ¯]([ÑÙ¯](a)) = 0 by 1.  .]
5. ¯=[Ñ¯]     º     ¯[Ñ¯]¯     =     0        [ HS 4-2.45 ]
[ Proof : ¯([Ñ¯](¯(a))) = [Ñ¯](^(¯(a))) = [Ñ¯](0) = 0  .]
^=[Ñ¯]     º     ^[Ñ¯]^     =     0
[ Proof : ^[Ñ¯]^ = ^¯[Ñ¯] = 0  .]
6. ¯Ñ(a,b) = ¯(a)×[ÑÙ¯](b) - ¯( a×[ÑÙ¯](b) ) = ¯(a)×[Ñ¯](b) - ¯( a×[Ñ¯](b) )
is crucially important because it expresses ¯Ñ geometrically in terms of [ÑÙ¯] and ¯ .
[ Proof : See [ HS 4-2.33 ].  .]
In particular, it gives ¯Ñ(¯(a),b) = ¯(a)×[ÑÙ¯](b) = ¯(a)×[Ñ¯](b) .
[ Proof : The projected product rule gives ¯(¯(a)[ÑÙ¯](b)) = ¯(a)¯([ÑÙ¯](b)) = 0 and as ¯(¯(a)Ù[ÑÙ¯](b)) = ¯(a)Ù¯([ÑÙ¯](b)) = 0 we must have ¯( ¯(a)×[ÑÙ¯](b) ) = 0  .]
7. [ÑÙ¯](b) = Ip-1¯Ñ(Ip,b) = Ip(ÐbIp-1 = -¯Ñ(Ip,b)Ip-1       expresses [ÑÙ¯] geometrically in terms of ¯Ñ and Ip without further differention and provides perhaps the simplest mental picture of the curl tensor as the bivector which acting geometrically on Ip returns -ÐpaIp , the negated rate of change of Ip in direction a. .
[ Proof :  a=Ip in 6 gives ¯Ñ(Ip,b) = Ip×[ÑÙ¯](b) = - [ÑÙ¯](b)Ip by 4.  .]
8. ¯Ñ(a,b)=¯(a)¿[ÑÙ¯](b) - ¯(a¿[ÑÙ¯](b))     [ HS 4-2.27 ]
[ Proof :  a×b<2> = a¿b<2> combined with 5  .]
9. (¯Ñ)2(Fp,a,b) =   ¯( [Ñ¯](b)×(¯(Fp)×[Ñ¯](a)) ) - ¯( Fp×[Ñ¯](a) )×[Ñ¯](b)
[ Proof :  Applying Shape Property 6 twice while exploiting Shape Property 3
¯Ñ(¯Ñ(Fp,a),b) =   ¯(¯Ñ(Fp,a))×[Ñ¯](b) - ¯( ¯Ñ(Fp,a)×[Ñ¯](b) )
= ¯(¯(Fp)×[Ñ¯](a) - ¯( Fp×[Ñ¯](a) ))×[Ñ¯](b)     -     ¯( (¯(Fp)×[Ñ¯](a) - ¯( Fp×[Ñ¯](a) ))×[Ñ¯](b) )
= 0 - ¯2( Fp×[Ñ¯](a) ))×[Ñ¯](b) - ¯( (¯(Fp)×[Ñ¯](a) )×[Ñ¯](b) )   + 0     by the projected bivector commutation rule  .]
If ¯(Fp)=Fp this reduces to
¯Ñ(¯Ñ(Fp,a),b) =   ¯( [Ñ¯](b)×(Fp×[Ñ¯](a)) )     [ HS 5-1.3 ]   which we can abbrviate to
¯Ñ2¯ = - ¯[Ñ¯]×[Ñ¯]×¯ .
10. [Ñ¯](Ip) = ÑpÙIp = ÑpIp     [ HS 4-2.44 ]
12. ¯Ip(b)¿[ÑÙ¯](a) = ¯Ip(a)¿[ÑÙ¯](b)     [ HS 4-2.28 ]
13. ¯ÑIp(a,d) = ¯Ip(b) ¿ [ÑÙ¯](d) - ¯Ip(a¿[ÑÙ¯](d))     [ HS 4-2.27 ]
14. ¯ÑIp([ÑÙ¯](a),d) = - ¯(Ip)([ÑÙ¯](a)×[ÑÙ¯](d))
16. a*(¯IpÑp(d,b)) = b*(¯IpÑp(d,a))
17. ¯IpÑ(d,aÙb) = ¯IpÑ(d,a) Ù ¯Ip(b)   +   ¯Ip(b) Ù ¯IpÑp(d,b)
18. ¯IpÑp(b,¯Ip(a)) = ¯IpÑp(a,¯Ip(b))
19. --- end list

[Ñ¯]  = ÑÞ ¯Ñ enables us to grade extend [Ñ¯] from the grade extension of_prl0g() as
[Ñ¯](aÙb) = Ñc¯Ñ(aÙb,c) = Ñc(¯Ñ(a,c)Ù¯(b)+¯(a)Ù¯Ñ(b,c))     so we immediately have [Ñ¯](^(a)Ù^(b))=0     [ HS 4-2.40c ].  Furthermore
[Ñ¯](¯(aj)Ù¯(b)) = ([Ñ¯](¯(aj))Ù¯(b) + (-1)j¯(aj)Ù([Ñ¯]¯(b))     [ HS 4-2.40a ]
[ Proof :  [Ñ¯](¯(aj)Ù¯(bk)) =  [Ñ¯]¯(ajÙbk) =  [ÑÙ¯](ajÙbk) = ÑcÙ( ¯Ñ(¯(aj),c)Ù¯(bk) + (¯(aj)Ù¯Ñ(¯(bk),c) )
= ÑcÙ( ¯Ñ(¯(aj),c)Ù¯(bk) + (-1)jk¯Ñ(¯(bk),c)Ù¯(aj) ) = (ÑcÙ¯Ñ(¯(aj),c))Ù¯(bk) + (-1)_jk(ÑcÙ¯Ñ(¯(b),c))Ù¯(aj)
= ([ÑÙ¯](¯(aj))Ù¯(bk) + (-1)jk([ÑÙ¯]¯(b))Ù¯(aj) = ([ÑÙ¯](¯(aj))Ù¯(bk) + (-1)jk+j(k+1)¯(aj)Ù[ÑÙ¯]¯(bk)
= ([Ñ¯](¯(aj))Ù¯(bk) + (-1)j¯(aj)Ù([Ñ¯]¯(bk))  .]

In particular, [Ñ¯]¯(aÙb)  = [Ñ¯]¯(a)Ù¯(b)-[Ñ¯]¯(b)Ù¯(a) .

[Ñ¯](^(aj)Ùb) =   [Ñ¯](^(aj)Ù¯(b)) =   [Ñ¯](^(aj))Ù¯(b) + (-1)j¯Ñ(^(aj))Ù(Ñ¿b)     [ HS 4-2.40b ]
[ Proof :  [Ñ¯](^(aj)Ù¯(bk)) = Ñc¯Ñ(^(aj)Ù¯(bk) ,c) = Ñc(¯Ñ(^(aj),c)Ù¯(bk)) = Ñc¿(¯Ñ(^(aj),c)Ù¯(bk)) + (ÑcÙ¯Ñ(^(aj),c))Ù¯(bk)
= (Ñc.¯Ñ(^(aj),c))Ù¯(bk) + (-1)j ¯Ñ(^(aj),c)(Ñc.¯(bk)) + [ÑÙ¯](^(aj))Ù¯(bk)     by the expanded inner product rule
= ([Ñ¯]^(aj) + (-1)j ¯Ñ(^(aj),c)(Ñc.¯(bk)) + 0  .]

====

The primary 1-differential of the extended curl (2;1)-tensor is
[ÑÙ¯]Ñ(a,b) º Ð¯(b) [ÑÙ¯](ap) - [ÑÙ¯]( Ð¯(b)(ap))  .

Hypercurve
For M=N-1 we have [ÑÙ¯](a) = np2 npnpÑ(a) = np2 npÙnpÑ(a)
[ Proof :  ÐbiN-1p = Ðb(npi) = (Ðbnp)i = npÑ(b)i so Shape Property 7 gives
[ÑÙ¯](a) = iN-1p-1( Ð¯(a)iN-1p) = i-1np-1npÑ(a)i = np2 _psiinvd(npnpÑ(a)) = np2 _psiinvd(npÙnpÑ(a)) = np2 npÙnpÑ(a)     since i commutes with all bivectors.  .]

[¯(Ñ)] = np2 np(Ñp¿np) = np2 np(Ñp¿np)
[ Proof :   Ña[ÑÙ¯](a) = -np2 Ña[npÑ(a)np] = -np2 (ÑanpÑ(a))np = -np2 (ÑaÐanp)np = -np2 (Ñpnp)np = -np2 np(Ñp¿np)  .]
Thus [Ñ¿¯](a) = [¯(Ñ)].a = -np2 (Ñ¿np)(ap¿np)

Squape 1-multitensor     [Ñ¯]2

The shape tensor [Ñ¯] raises the grade of ¯Ip(a) while lowering the grade of ^Ip(a), preserving neither grade nor tangency (containment within Ip) but the squared shape or squape 1-tensor [Ñ¯]2 preserves both with [Ñ¯]2  = ^=([Ñ¯]2)  + ¯=([Ñ¯]2) .
[ Proof :   Follows from ¯=[Ñ¯] = ^=[Ñ¯] = 0 and [Ñ¯]^ = ¯[Ñ¯] since
[Ñ¯]2 = (¯+^)[Ñ¯](¯+^)[Ñ¯](¯+^) = ^[Ñ¯]¯[Ñ¯]^  + ¯[Ñ¯]^[Ñ¯]¯ = ^[Ñ¯]2^2  + ¯[Ñ¯]2¯  .]

We also have [Ñ¯]2^ = ^[Ñ¯]2   = [Ñ¯]¯[Ñ¯] º [Ñ¯]=¯ . [ HS 4-2.46b ]

The squape tensor decomposes as [Ñ¯]<2> + [Ñ¯]<1> + [Ñ¯]<0>

Ricci 1-tensor       ([Ñ¯]2¯)
The intrinsic squape 1-multitensor is the projection of the squape ¯[Ñ¯]2   = [Ñ¯]2¯ =   ¯=([Ñ¯]2) , acting entirely upon and within Ip .
We refer to the instrinsic squape acting only on 1-vectors as the Ricci 1-tensor.

We will show eventually that
[Ñ¯]2 ¯   =   (ÑÙÑ) ¯   =   (ÑÙÑ) ×¯     [ HS 5-1.28 5-1.29 ] abbreviating
[Ñ¯]2(¯(ap)) = (ÑÙÑ)(¯(ap)) = (ÑÙÑ)×(¯(ap))     " ap .

[Ñ¯]2¯(aÙb) = [Ñ¯]2(a) Ùb + 2¯( [Ñ¯]Ñ(b)×[Ñ¯]Ñ(a) ) + aÙ [Ñ¯]2(b)     [ HS 4-2.48 ]
[ Proof :  ... ???  .]

¯[Ñ¯]2 = -Ñ[ÑÙ¯] = -Ñ¿[ÑÙ¯]     abbreviating ¯[Ñ¯]2(b) = -Ñp[ÑÙ¯](b) = -Ñp¿[ÑÙ¯](b) . [ HS 5-1.19 ]
[ Proof : Ña   ¯[Ñ¯]×(a,b) = -ÑaÐa[ÑÙ¯](b) = -Ñp[ÑÙ¯](b)     by Curvature Identity 2   below  .]

[ Proof : ???  .]

Curvature 2-tensor     [Ñ¯]×
Recall that the commutator product of two bivectors is itself a bivector.
If Ip satisfies the integrability condition then the antisymmetric full curvature 2-tensor
cpaÙb º [Ñ¯]×(a,b) º [Ñ¯](a)×[Ñ¯](b) = [ÑÙ¯](a)×[ÑÙ¯](b)

satisfies [Ñ¯]×(a,b) = ½([ÑÙ¯]Ñ(a,b) - [ÑÙ¯]Ñ(b,a))     [ HS 4-4.17 ] where [ÑÙ¯]Ñ(ap,b) º Ð¯(b)([ÑÙ¯](ap)) - [ÑÙ¯]( Ð¯(b)ap) .
[ Proof :  ?????    See HS  .]
We can write this with regard to 1-vector arguments as
[Ñ¯]×   =   [ÑÙ¯]×   =   ([ÑÙ¯]Ñ)×     º     [ÑÙ¯]Ñ×
Thus we can categorise some second dervivative properties of an M-curve geometrically from the first derivative shape.

Curvature differential (2;3)-tensor     [Ñ¯]×Ñ

This enables us to express the primary differential of the curvature:
[Ñ¯]×Ñ(a,b,c) º ([Ñ¯]×(a,b))Ñ(c) º Ðßc([Ñ¯]×(a,b)) º (Ðc([Ñ¯]×))(a,b)
as ½([ÑÙ¯]Ñ(a,b,c)-[ÑÙ¯]Ñ2(b,a,c)) = ½([ÑÙ¯]Ñ(a,b,c)-[ÑÙ¯]Ñ2(b,c,a)) .
Cyclically permuting the a,b,c and summing we obtain the generalised Bianchi identity
Sabc ([ÑÙ¯](a)×[ÑÙ¯](b))Ñ(c)  = 0
Sabc ([Ñ¯]×)Ñ(a,b,c)  º  Sabc Ðßc([Ñ¯]×)(a,b)   =   0     [ HS 5-1.39 ] which we can also write as ([Ñ¯]×)Ñ¿ = 0 . The curvature thus has vanishing exterior 1-differential.

Setting a=[Ñ¯](a) in Projection Property 6 [ [ HS 4-2.33 ]] gives
¯Ñ([Ñ¯](a),b) = ¯([Ñ¯](a))×[Ñ¯](b) - ¯([Ñ¯](a)×[Ñ¯](b)) = - ¯([Ñ¯](a)×[Ñ¯](b)) º  - ¯([Ñ¯]×(a,b))
which we can abbreviate (with 1-vector arguments understood) to
¯ [Ñ¯]×   =   ¯ [ÑÙ¯]×   =   -¯Ñ [ÑÙ¯]
ie.   1Ñ2 [ÑÙ¯] = - 1Ñ [ÑÙ¯]×  = - 1Ñ [Ñ¯]× .

Since the curvature 2-tensor is skewsymmetric and bilinear in its two nonprimary arquments it defines a bivector-valued curvature (*1)-multitensor] of a single bivector argument
[Ñ¯]×(aÙb) º [Ñ¯]×(a,b) . It is natural to extend this to a multivector argument via [Ñ¯]×(a) = [Ñ¯]×(aÙ1) = [Ñ¯]×(a,1) = 0 . º [Ñ¯]×(a,b) . It is natural to extend this to a multivector argument via [Ñ¯]×(a)=[Ñ¯]×(a)=0

Intrinsic curvature 2-tensor       ¯[Ñ¯]×

The intrinsic curvature of an M-curve can be defined in a number of ultimately equivalent ways. We will here regard it primarily as the projection of the full 2-curvature
¯( [Ñ¯](a)×[Ñ¯](b)) = ¯([ÑÙ¯](a)×[ÑÙ¯](b) )     and denote it by
¯[Ñ¯]×     =     ¯[ÑÙ¯]×
with ¯[Ñ¯]×(a,b) º ¯( [Ñ¯](a)×[Ñ¯](b) ) .
It is bivector-valued and bilinear in a,b and is thus an antisymmetric 2-tensor, ie. a 2-form. The remaining, rejected, component of the full curvature existing outside Ip is known as the extrinsic curvature 2-tensor.

Riemann curvature 2-tensor       ¯=[Ñ¯]×

The Riemann curvature of an M-curve is the intrinsic curvature restricted to the M-curve
¯=[Ñ¯]×(a,b) º ¯[Ñ¯]×(¯(a),¯(b)) = ¯([Ñ¯](¯a)×[Ñ¯](¯ |b)) = ¯([ÑÙ¯](a)×[ÑÙ¯](b)) = ¯[Ñ¯]×(¯(aÙb))
so that nonprimary arguments outside Ip are mapped to 0.

(¯[Ñ¯]×)Ñ¿(a,b,c) = Sabc ¯([Ñ¯]×^(aÙb))×[Ñ¯]¯(c)     with the immediate consequence (¯=[Ñ¯]×)Ñ¿(a,b,c) = 0 , ie. the Riemann curvature has vanishing exterior 1-codifferential, which is the traditional second Bianchi identity.
[ Proof :  Ðßc ¯[Ñ¯]×(aÙb) º (Ðc ¯[Ñ¯]×)(aÙb) = ¯Ñ , c)) + ¯( Ðßc[Ñ¯]×(aÙb))
Þ Sabc (Ðc ¯[Ñ¯]×)(aÙb) = Sabc ¯([Ñ¯]×(aÙb))×[Ñ¯](c) .     Hence
Sabc (Ðc ¯[Ñ¯]×)(aÙb) = Sabc ¯Ñ , c)  + ¯( Sabc Ðßc[Ñ¯]×(aÙb)) = Sabc ¯Ñ , c)  + ¯( 0)
= Sabc (¯([Ñ¯]×(aÙb))×[Ñ¯](c)   - ¯([Ñ¯]×(aÙb) ×[Ñ¯](c) )       by Shape Proprety 6
= Sabc ¯([Ñ¯]×(aÙb))×[Ñ¯](c)   - ¯(Sabc (([Ñ¯](a)×[Ñ¯](b))×[Ñ¯](c) ) = Sabc ¯([Ñ¯]×(aÙb))×[Ñ¯](c)       by Jacobi Identity .  .]

Hence Sabc (Ðc ¯[Ñ¯]×)(aÙb) = Sabc ¯( Ð¯(c) ¯[Ñ¯]×)(aÙb) ) = Sabc ¯( ¯([Ñ¯]×(aÙb))×[Ñ¯](¯(c)) )
= Sabc ¯( [Ñ¯]×(aÙb))×[Ñ¯](¯(c)) )        by the projected bivector commutation rule
Replacing (aÙb) with ¯(aÙb) gives a vanishing result by the generalised Bianchi identity.  .]

We can express the Riemann curvature  ¯( [Ñ¯](a)×[Ñ¯](b)) solely in terms of the intrinsic squape multitensor via
¯[Ñ¯]×(aÙb) = ½( [Ñ¯]2(a)Ùb + aÙ[Ñ¯]2(b) - [Ñ¯]2(aÙb) )     " a,b Î Ip     [ HS 4-2.48 ].

Since [Ñ¯]2 preserves grade we must have
Ña¿[Ñ¯]2(aÙb) = -2Ña¿(¯[Ñ¯]×(a,b)) = -2¯(Ña)¿([Ñ¯]×(a,b)) = -2Ña¿([Ñ¯]×(a,b))     and similarly
Ña¿[Ñ¯]2(aÙb) = -2Ña¿(¯[Ñ¯]×(a,b)) = [Ñ¯]2(b) .

The Riemann curvature is protractionless
ÑaÙ¯[Ñ¯]×(¯(aÙb))   =   0     [ HS 5-1.11 ] .

We also have
ÑßÙ¯[Ñ¯]×¯   =   0     [ HS 5-1.13a ]
[ Proof : See [ HS p191 ]  .]

There are various alternate and equivalent definitions for the intrinsic 2-curvature one can adopt based on the following Curvature Identites.

1. ¯[Ñ¯]×   =   -¯Ñ[ÑÙ¯]     abbreviating ¯( [ÑÙ¯]×(a,b) ) = -¯Ñ([ÑÙ¯](a),b) = ¯Ñ([ÑÙ¯](b),a)     [ HS 5-1.5b ]
[ Proof :  Substituting ap=[ÑÙ¯](a) into  Shape Property 6   and exploiting Shape Property 3 :
¯Ñ([ÑÙ¯](a),b) = ¯([ÑÙ¯](a))×[ÑÙ¯](b) - ¯( [ÑÙ¯](a)×[ÑÙ¯](b) ) = 0 -¯( [Ñ¯](a)×[Ñ¯](b) )  .]
2. ¯[Ñ¯]×   =   [ÑÙ¯]Ñ = (Ð [ÑÙ¯])     abbreviating ¯[Ñ¯]×(a,b) = (Ðb[ÑÙ¯])(a) º Ðb([ÑÙ¯](a))-[ÑÙ¯](Ðba)     [ HS 5-1.5c ]
[ Proof :  (Ðb[ÑÙ¯])(a) = Ð¯(b)(¯([ÑÙ¯](a))) - ¯Ñ([ÑÙ¯](a),b) = Ð¯(b)0 - ¯Ñ([ÑÙ¯](a),b) and result follows from 1.  .]
3. ¯[Ñ¯]××¯    =   -2(¯Ñ)2×¯     abbreviating ¯([Ñ¯]×(a,b))×¯(Fp) =  2(¯Ñ(¯Ñ(¯Fp,a),b)-¯Ñ(¯Ñ(¯Fp,b,a))     [ HS 5-1.4 ]
[ Proof : Shape Property 9 gives
¯Ñ(¯Ñ(¯Fp,a),b) - ¯Ñ(¯Ñ(¯Fp,b),a) =   ¯( [Ñ¯](b)×(¯(Fp)×[Ñ¯](a)) ) -   ¯( [Ñ¯](a)×(¯(Fp)×[Ñ¯](b)) )
=   ¯( ¯(Fp)×([Ñ¯](b)×[Ñ¯](a)))     by Jacobi identity
=    ¯(Fp)×¯([Ñ¯](b)×[Ñ¯](a)))     by projected bivector commutation rule  .]
4. ¯[Ñ¯]××¯   =   -2Ð×¯     abbreviating ¯([Ñ¯]×(a,b))×¯(Fp) = - 2(Ða×Ðb)¯(Fp)     [ HS 5-1.4 ]
[ Proof :   Follows immediately from (¯Ñ)2×¯ = Ð ×¯ and the previous identity.  .]
cpab)×Fp º -(Ða×Ðb)Fp with FpÎIp understood is the traditional approach in much of the literature. For functions satisfying the integrability condition, directed derivatives commute but directed coderivatives may not. In restricting Ñp to act "within and withon" Ip, we "break" the commutivity of directed derivatives, but the commutativity "survives" both in the form of the antisymmetry of the Riemann 2-curvature so defined and also in the Ricci Identity
ÑÙÑÙ b2p = 0 for any bivector field b2p
This enables us to also regard the Riemann curvature as a pointer-dependant bivector-specific 1-tensor
cpaÙbc) º cpaÙb×c which we shall see later has the geometric interpretation of the 1-vector change in c if it is "paralell transported" around (the projection into CM of) a tiny planar loop through p in aÙb ,  divided by the content of that loop.

It is easy to fail to appreciate its true significance of this. In consequence of the integrability condition of Ip, it is possible to evaluate (Ða×Ðb)Fp without having to differentiate Fp by applying a particular linear function independant of Fp to Fp. This is why ÑÙÑ is essentially geometric, with no differentiating component.

5. ¯[Ñ¯]2   =   ÑÞ ¯[Ñ¯]×   =   ÑÞ ¿ ¯[Ñ¯]×   =   ÑÞ ¿ ¯[Ñ¯]×     provides the more traditional definition of the Rici 1-tensor as the contraction of the Riemann curvature 2-tensor . [ HS 5-1.17 ]
[ Proof :  ¯[Ñ¯]([Ñ¯](b)) = [Ñ¯]([Ñ¯](¯b) ) = [Ñ¯]([ÑÙ¯](b)) = (ÑaÐpa¯)([ÑÙ¯](b)) = Ña¯Ñ([ÑÙ¯](b),a) = Ña¿¯Ñ([ÑÙ¯](b),a) combined with Curvature Identity 1 . Finally, ÑÞ¿¦p(ap) = ¯(ÑÞ¿¦p(ap)) = ÑÞ¿¦p(ap) for any 1-vector-valued ¦p(ap) .  .]
6. ¯=[Ñ¯]×(aÙb)   =   (b¿Ñp) [ÑÙ¯](a)   =   Ðb[ÑÙ¯](a)     [ HS 5-1.5c ]
expresses ¯=[Ñ¯]×(aÙb) as the b-directed primary coderivative of [ÑÙ¯](a), the geometric operator that sends Ip to ÐpaIp .
[ Proof :  ???  .]
7. ¯=[Ñ¯]×(aÙb)   =   ¯ÑIp([ÑÙ¯](b),a)   =   ÑuÙ¯ÑaÑv ¯Ñ(¯Ñ(v,b),a)     [ HS 5-1.5b ] .
[ Proof :  ???  .]
8. ¯=[Ñ¯]×(aÙb)   =    (ÑuÙÑv)(¯Ña(u)¿¯Ñb(v))     [ HS 5-1.5d ] .
[ Proof :  ???  .]
9. ÑßÙ¯=[Ñ¯]×    =   0 abbreviating (ÑÙ¯=[Ñ¯]×)(aÙb)=0     [ HS 5-1.13a ] .
[ Proof :  ???  .]
10. ¯=[Ñ¯]×(ÑÜÙb)   =   ÑÙ([Ñ¯]2¯)(b)     [ HS 5-1.22 ].
[ Proof : ¯=[Ñ¯]×Ñ(ÑÙb) º ¯(¯=[Ñ¯]×Ñ(ÑÙb)) = ¯((ÑÞ¿Ñ)¯=[Ñ¯]×(aÙb))     by the tangential gradifying substition rule
= ¯( ÑÞ¿(ÑÙ¯=[Ñ¯]×(aÙb) + ÑÙ(ÑÞ.¯=[Ñ¯]×(aÙb)) )     via a¿(bÙc)=(a.b)c-bÙ(a.c)
= ¯( Ñ¿0 + ÑÙ([Ñ¯]2¯)(b)) ) º ÑÙ([Ñ¯]2¯)(b)     by the Bianchi identity.  .]

We thus have ÑßÙ[Ñ¯]×(aÙb) º (ÑÙ[Ñ¯]×¯)(aÙb) = ÑaÙ¯ÑIp([ÑÙ¯](b),a) = 0     which we can write as the Bianchi identity
ÑßÙ¯=[Ñ¯]× = 0     [ HS 5-1.13a ].
[ Proof :   (Ðc¯=[Ñ¯]×)(aÙb) = Ðc( Ð¯(v)Ù The gradifying substitutuin rule gives _..... ?????  .]

Hypercurve
For M=N-1 we have [Ñ¯]×(a,b) = ¯[Ñ¯]×(a,b) = - npÑ(aÙb) .
[ Proof :  [Ñ¯](a)×[Ñ¯](b) = [ÑÙ¯](a)×[ÑÙ¯](b) = np2 (npnpÑ(a))×(npnpÑ(b)) = -np4 (npÑ(a)×npÑ(b)) = - npÑ(a)ÙnpÑ(b) º - npÑ(aÙb)  .]
Hence [Ñ¯]2¯(b) = Ña¿¯[Ñ¯]×(a,b) = -Ña¿(npÑ(a)ÙnpÑ(b)) = ???

Scalar Curvature (0;1)-tensor         R = ÑÞ [Ñ¯]2¯

The scalar curvature (aka. total curvature) is traditionally presented as the second contraction of the Rieman curvature     R º (ÑÞ¿)2 ¯[ÑÙ¯]×     =   ÑÞ2 ¯[ÑÙ¯]×     =   ÑÞ [Ñ¯]2¯ ;
but is also, perhaps more fundamentally, the tangential point divergence of the spur     Ñ¿[¯(Ñ)]     [ HS 5-1.21 ] .

Ñp¿[Ñ¯]2¯(a) =  Ñp¿[Ñ¯]2¯(a) = Ñp¿[Ñ¯]2¯(a)

2 [Ñ¯]2Ñ(Ñ)   =   ÑR     [ HS 5-1.23 ]
[ Proof : Contracting the contracted Bianchi identity yields
Ñb¿((ÑÙ([Ñ¯]2¯))(b)) = Ñb¿(¯=[Ñ¯]×Ñ(ÑÙb)) Þ (Ñb¿Ñ)([Ñ¯]2¯)Ñ(b) - Ñ(Ñb¿([Ñ¯]2¯)Ñ(b)) = - Ñb¿(¯=[Ñ¯]×Ñ(bÙÑ))
Þ ([Ñ¯]2¯)Ñ(Ñ)) - Ñ(RÑ) = -([Ñ¯]2¯)Ñ(Ñ))     Þ 2([Ñ¯]2¯)Ñ(Ñ)) = Ñ(RÑ) .  .]

Einstein 1-tensor         (1-½ÑÞ)[Ñ¯]2¯
2([Ñ¯]2¯)Ñ(Ñ)) = Ñ(RÑ)  means that the symmetric (self-adjoint) Einstein 1-tensor
(1-½ÑÞ)[Ñ¯]2¯(a)   =   ([Ñ¯]2¯)(a) - ½Ra   =   [Ñ¯]2¯(a) - (Ñ¿[¯(Ñ)])a
has (1-½ÑÞ)[Ñ¯]2¯Ñ(Ñ) = ¯(1-½ÑÞ)[Ñ¯]2¯Ñ(Ñ) = 0 and so is zero (point) codivergent, ie.
[Ñ¿(1-½ÑÞ)[Ñ¯]2¯](a)   =   [Ñ¿(1-½ÑÞ)[Ñ¯]2¯](a)   =   0     " a Î Ip . [ HS 5-1.23 ]
It's (directional) cocontraction is Ña¿((1-½ÑÞ)[Ñ¯]2¯(a)) = Ña¿((1-½ÑÞ)[Ñ¯]2¯(a)) = ½(2-M)R .

(1-½Ña)(ÑpÙÑp) = 0

Next : The Coordintae Based Approach

Glossary   Contents   Author
Copyright (c) Ian C G Bell 1998
Web Source: www.iancgbell.clara.net/maths or www.bigfoot.com/~iancgbell/maths
Latest Edit: 18 May 2007.