By defining an

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

*M*-field *I*_{p} can be expressed in *U*_{N} via ^{N}*C*_{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 *N*^{2} 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 ^{M}*C*_{2}^{2} = ¼*M*^{2}(*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

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

c

The mapspace representation of the Ricci 1-tensor consists of

The mapspace Einstein tensor is given by

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

Such paths are called

It is important to note that the 1-curve streamlines of field

Suppose now we have *M* not-necessarily mutually orthogonal unit 1-fields **u _{i p}** defined over

An obvious example is

Given a general multivector-valued function F(**p**) defined over ** C_{M}** ,

If we so differentiate the identity map F(

We introduce the notation _{ Dli}¬^{i }**p** to indicate the point **p _{i}**(

We can extend

This defintion is integrative, good for all

We might be tempted to try to parameterise ** C_{M}** using

[ That is, starting from a fixed reference point

Any

Only if the

[ Proof : ... .]

(_{ e}¬^{i } __×__ _{ e}¬^{j }) is a 1-field defined over ** C_{M}** returning
a

Taking e®0 we can visualise (

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.

The

[

which we can write as [

The use of

Any one of our

For

It

Though

We refer to (

We define the Lie drag of a scalar field f(

The 1-vector Lie derivative of a scalar field is simply

Since the Lie drag of **u _{jp}** (or

(

[ Proof :

Hence

Omitting the brackets , we thus have the 1-vector operator identity

Let us now identify the spaces

[ Proof :

For brevity we ommit an

Any "inner product"

For ¿ in Â

With regard to

In a Euclidean space,

(1;0)-tensor

(0;1)-tensor

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 **q**Î** C_{M}**
a function

Let ^{i}^{Dli}_{p0}(**a**) denote the parallel transport of **a** from **p _{0}** for arclength

We obviously want

The normalisation condition for ^{i}^{Dli}_{p}(**a**) means that
the right-connection
b_{a,p}(*D*l_{i}) º a^{-1}^{i}^{Dli}_{p}(**a**)
for ^{i}^{Dli} at **p** satisfies
b_{a,p}b_{a,p}^{§} = 1 and is thus a 1-rotor expressable as a 2-spinor *e*^{wui p a p(Dli)} ,
where w_{ui p a p}(*D*l_{i})
= *cos*^{-1}(**a**^{-1}¿^{i}^{Dli}_{p}(**a**) (**a**^{-1}Ù^{i}^{Dli}_{p}(**a**)^{~}
is a *U*_{N} 2-blade.
[ we assume *D*l_{i} is sufficiently small and ** C_{M}** 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

where for given scalar e, w

Parallel displacement along the

**g**_{p}(**u _{jp}**,

is a 1-vector lying in general neither in

We make the continuity assumption that

Taking

A *linear connection* (*aka.* *affine connection*) derives from an assumed
bilinearity of **g**_{p} which can then
can be fully defined by *M*^{2} 1-fields
**g**_{p}( **h**_{jp }, **h**_{ip })
which then effectively define _{q}^{p C}(**a**).
Indeed, some authors define ^{i}^{Dli} as the consequence of a given linear connection.

We say the connection is symmetric if
**g**_{p}(**u _{i p}**,

Suppose now that the **u _{i p}** are everywhere mutually orthogonal so that {

The connection is symmetric if

A linear connection arises if w

[ Proof : Since the

= (å

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**__¿___{x}**b**
.]

Let **b** be a map direction at **x**Î** M_{ap}** with corresponding

is, for given

G

[ 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.

The directed coderivative of

[ Proof :

=

In the limit, since

Accordingly we can define a directed coderivative operator

for a given direction

We then have

[ Proof :

__Ð___{a} can be regarded as a differential over and "within" ** C_{M}**, although its component parts
Ð

__Ð___{a} generates a 1-vector *codel-operator*

__Ñ___{p} º å_{k=1}^{M} **u ^{k}_{p}**

Within the mapspace

__Ð___{a} induces a differential operator within *M*-D parameter space ** M_{ap}**Ì

In particular,

We say a 1-field

where

The physical interpretation of geodesics are the potential trajectories of "free falling" particles and (assuming a natural streamline parameterisation

derived from

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

[Under Construction]

The traditional

For a 3D 2-curve we have tangent unit vectors

Rotation about

Their product k

If the surface is parameterised as **p**(l_{1},l_{2})
with **h**_{1p } = ¶**p**/¶l_{1} , **h**_{2p } = ¶**p**/¶l_{2}
then

¶^{2}**p**/¶*l _{i}*¶l

¶

2-curvature is geometrically representable as ortho bi-circle S_{1}+S_{2} for 3-blade osculating at **p** principle 1-spheres S_{1} and S_{2}.
Their product S_{2}S_{1} embodies the rotation taking S_{1} into S_{2},
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 k_{1}=k_{2} we have instead of 1-spheres a 4-blade 2-sphere for umbilical **p** or
a 4-blade 2-plane when k_{1}=k_{2}=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 k_{1}^{-1}, k_{2}^{-1},... of the principle curvatures
and for *M*³2 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
**E**º**E** 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}**,

which is clearly dependant solely on value of

We thus have the geometric interpretation of

[ Proof :

The *second directional coderivative* operator
__Ð___{li}__Ð___{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 :

.]
__Ð___{b}__Ð___{a}**u** =
( Ð _{¯(b)} + w_{b}__×__)( Ð_{¯(a)} + w_{a}__×__)**u**
=
( Ð _{¯(b)} + w_{b}__×__)( Ð_{¯(a)}**u** + ½(w_{a}**u**-**u**w_{a}))
= ( Ð_{¯(b)} Ð_{¯(a)}**u** + ½( Ð_{¯(b)}w_{a}**u**- Ð_{¯(b)}**u**w_{a})
+w_{b}__×__( Ð_{¯(a)}**u** + ½(w_{a}**u**-**u**w_{a}))
=
Ð _{¯(b)} Ð_{¯(a)}**u** + ½( Ð_{¯(b)}w_{a}**u** - Ð_{¯(b)}**u**w_{a}
+ w_{b} Ð_{¯(a)}**u** - Ð_{¯(a)}**u**w_{b} )
+ ½w_{b}__×__(w_{a}**u**-**u**w_{a})
=
Ð _{¯(b)} Ð_{¯(a)}**u** +
½( Ð_{¯(b)}(w_{a}**u**) + w_{b} Ð_{¯(a)}**u**
- Ð_{¯(a)}(**u**w_{b})
- Ð_{¯(b)}(**u**w_{a}) )
+ ¼(
w_{b}w_{a}**u** - w_{b}**u**w_{a}
- w_{a}**u**w_{b} + **u**w_{a}w_{b} )
Þ 2(¶ _{b}__×__¶_{a})**u** =
2( Ð _{¯(b)}__×__ Ð_{¯(a)} + Ð_{¯(b)}__×__(w_{a}__×__) + (w_{b}__×__)__×__ Ð_{¯(a)} +
(w_{b}__×__)__×__(w_{a}__×__))**u**
=
Ð _{¯(b)}(w_{a}**u**)- Ð_{¯(a)}(w_{b}**u**) + w_{b} Ð_{¯(a)}**u** -w_{a} Ð_{¯(b)}**u**
+ (w_{b}__×__w_{a})__×__**u**
= ( Ð_{¯(b)}w_{a})**u** - ( Ð_{¯(a)}w_{b})**u** + (w_{b}__×__w_{a})__×__**u**
=
( Ð _{¯(b)}w_{a})__×__**u** - ( Ð_{¯(a)}w_{b})__×__**u** + (w_{b}__×__w_{a})__×__**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

It is possible to express C

The last of these reflects the reversability of parallel transport.

We also have (from the G expression for

which can also be expressed as C

It follows that

As a result, the

__Ð___{ei}^{ð}C_{xejk} + __Ð___{ej}^{ð}C_{xeki} + __Ð___{ek}^{ð}C_{xeij} = 0_{2}
is known as the *second Bianci identity*.

The codivergence of the Ricci tensor
__Ñ__^{ð}_{x}¿ c_{pa}
º å_{i}**e ^{i}**¿

[ Proof :

The differential operator

Suppose now that

[ Proof :

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]

×××××××××××××××××××××××××××××××××××××××

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 [

Their Ñ is our

Their ¶ is predominantly our

Their d

Their (

Their

Their g

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

Next : Multivector Physics

Glossary Contents Author

Copyright (c) Ian C G Bell 1998, 2014

Web Source: www.iancgbell.clara.net/maths

Latest Edit: 04 Oct 2014.