6 Levi-Civita Connection on a Surface and Associated CovariantDerivative

6.1 Section Overview and Summary

A summary of this section is as follows. Let S be a surface in R3 parameterized by the coordinate functionsqi = �i(u↵) (i = 1 . . . 3,↵ = 1 . . . 2) in terms of the (generalized or curvilinear) coordinates u↵. Each set ofcoordinates u↵ defines a set of coordinated vectors denoted @↵ that form a basis for the tangent plane to S

at any point. For each coordinate system u↵, assign to S the n(n2+n)2 (n=2) functions (called the Christo↵el

symbols (of the second kind) ) defined by

n

⌘↵�

o

�=

1

2g�⌘ (�g↵�,� + g↵�,� + g��,↵) . (42)

In eq. (42) g↵� is the function obtained by computing the dot product of @↵ with @� (denoted g(@↵, @�) orh@↵, @�i and where g↵�,� is the partial derivative of the function g↵� w.r.t. u�. This assignment is called a(classical) connection or more specifically the Christo↵el connection.

(On the Symmetry of the Christo↵el Symbols) By the symmetry of the inner product g↵� = g�↵

and by their definition, the Christo↵el symbols are symmetric in the lower indices. That is,n

�↵�

o

=n

��↵

o

which is why there are n(n2+n)2 functions instead of n

3.

(On Distributing Partial Di↵erentiation to Each Entry of the Inner Product g(·, ·)) The geo-metric derivation of eq. (42) hinges on the fact that the for each ↵,� the “acceleration” a↵� = @� [@↵] splits

into a tangential componentn

⌘�↵

o

@⌘ and normal component S↵� n. Assuming that partial di↵erentiation

distributes across the entries of inner product12 as

�

=g↵�,�z }| {

@� [g(@↵, @�)] = ha↵� , @�i+ h@↵, a��i (43)

then the combination of g↵�,� + g��,↵ � g�↵,� can be solved for the Christo↵el symbols above.

The relationship between the Christo↵el symbolsn

�↵�

o

andn

ijk

o

in two di↵erent coordinate systems u↵

and ui is given by the transformation formula

�

kij

= �k,⌘�⌘,ij +

�

�↵�

�↵,i��,j�

k,� (44)

where �⌘,i is the ⌘-row and j-column entry of the Jacobian matrix of � where � is the change of coordinates

from u↵ to ui.

(On Coordinated Bases and Frobenius Theorem): The importance of this transformation law can beexplained physically (forces of constraint, etc), but take it for granted for the time being that the transfor-mation law is of the utmost importance. In the u↵ coordinate system on S, @↵ form a basis for the tangentspace to S. Switching to another basis of the tangent space, say vk defined by vk = M↵k @↵ where M

↵i is an

invertible matrix and following through with transform ofn

�↵�

o

to�

kij

one will find that because of some

asymmetries in derivatives of the matrix M , transformation equation eq. (44) cannot be obtained becausecertain terms will not cancel. Making the assumption that vk are coordinated vectors w.r.t. some coordinatesystem ui, these terms (cf. eq.(69)) cancel and the necessary tranformation law is obtained.

We are left with the conclussion that the Christo↵el symbols and its transformation law are defined andtrue only in a coordinated basis of the tangent space to S. The test to determine when a given set of vectorsvk is a coordinated set of vectors is given by the Frobenius theorem and says that if the commutatorbracket [vi, vj ] = 0 then the vectors are coordinated.

12see notational footnote10 about inner products entries that are (a) both tangent vectors to the surface and (b) where atleast one entry is a vector not in the surface.

18

Given the Christo↵el connectionn

�↵�

o

, a covariant derivative (w.r.t. this connection) is defined as13

r↵[@� ] =�

�↵�

@�. (45)

This definition is extended by endowing it with the Leibnitz properties

(L1) r� [M↵i @↵] = r� [M↵i ]@↵ +M↵i r� [@↵]= M↵i,�@↵ +M

↵� @↵;� (46)

(L2) rM�i @� [@↵] = M�i r� [@↵]. (47)

These properties are chosen so as to recover the transformation law of the Christo↵el symbols. With theseproperites covariant di↵erentiation r� [@↵] is extended to rX [Y ] and further to covectors rX [✓] and themetric tensor rX [g].

The implications of working in a coordinated basis are felt in terms of the torsion vector defined by

T (X,Y )�= [X,Y ]� (rX [Y ]�rY [X]) . (48)

When X = @i and Y = @j then

T (@i, @j) = [@i, @j ]���

kij

��

kji

�

@k = 0 + 0, (49)

and so there is no torsion for these given vectors and hence no torsion to the Christo↵el connection. In fact,for any vectors X and Y , the commutator terms cancel leaving only

T (X,Y ) = XiY jT (@i, @j) (50)

= XiY j�

[@i, @j ]��

kij

+�

kji

�

@k = 0, (51)

and so the torsion for any two vectors is 0. It is also shown that rX [g] = 0 (i.e. g is covariant constant) andtherefore, like partial di↵erentiation, covariant di↵erentiation distributes across the inner product as

rX [g(Y, Z)] = �g(rX [Y ], Z)� g(Y,rX [Z]). (52)

Conclussion: The Christo↵el connectionn

�↵�

o

defines a covariant derivative r↵[@� ] =n

�↵�

o

@� which

can be extended to covariant di↵erentiation rX [Y ] of vectors X = X↵@↵ and Y = Y �@� , by linearity andLeibnizian properties. With respect to this covariant di↵erentiation, the torsion vector T (X,Y ) (w.r.t. theChristo↵el connection) is zero. Further, the metric tensor g is also covariant constant, i.e. rX [g] = 0. As welook to move beyond/generalize the Christo↵el connection, realize that we can attempt to do so by addingin torsion14 and/or by dropping the covariant constant feature. The details of the Christo↵el (Levi-Civita)connection and associated covariant derivative are now given.

6.2 Tangential and Normal Decomposition of Acceleration: Geometric Mean-ing of Christo↵el Symbols

6.2.1 The Christo↵el Symbols of the First and Second Kind, {jk, i} andn

ijk

o

Consider a parameterized surface S embedded in Euclidean R3 with (generalized) coordinates u↵ and coor-dinate functions �i(u↵) = qi where i = 1 . . . 3 and ↵ = 1 . . . 2. By Euclidean R3, we mean R3 equipped withthe standard inner (dot) product h·, ·i given by hv, wi =

P3i=1 v

iwi. The (column) vectors

@↵ = �,↵ = (Jac�)↵ (53)

13Note that the directional covariant derivative index, ↵ in this case, goes into the first lower spot of the Christo↵el symbolswhile the vector index, � in this case, goes into the second lower spot. In other words, preserve the order r

v

1

[v2] = �312v3.While this ordering is not actually necessary for the Christo↵el connection, since the Christo↵el symbols are symmetric in theselower indices and is therefore a symmetric connection, the orderding will matter for the non-symmetric connections we willintroduce shorthly.

14Again, whatever this means. The n-symplectic approach to the frame dynamics should show these to be exter-nal/imposed/control forces on an orthonormal frame

19

define a basis for the tangent plane to the surface S at any point. Define the unit normal vector to thesurface as

n =@1 ⇥ @2k@1 ⇥ @2k

. (54)

where k · k =p

h·, ·i. For each entry of each vector @↵ denoted @i↵, we can compute the various accelerations

ai↵��= @i↵,� = �

i,↵� (55)

In other words, ai is the Hessian matrix of the ith coordinate function �i whose ↵th row and �th columnentry is ai↵� . We say that

a = [a1|a2|a3]t (56)

is the Hessian of �, Hess�. Unlike the vectors @↵ which lie in some tangent plane to S, the vectors

a↵� = @� [@↵]�= (@i↵),�)@i generally have a tangential and normal component. That is, for each i the Hessian

of �i can be decomposed intoai↵� = @

i�

�

�↵�

+ b↵� ni (57)

where the matrix of functions b↵� and the array of functionsn

�↵�

o

are still to be determined. We can do

the decomposition for all i at once by writing

a↵� = @� [@↵] = @��

�↵�

+ b↵� n. (58)

The functions b↵� andn

�↵�

o

can be found as follows. Using orthogonality of @↵ and n gives two equations

b↵� = ha↵� , ni (59)ha↵� , @⌘i = g�⌘

�

�↵�

(60)

where g�⌘ is the matrix of functions defined by g(@�, @⌘) = h@�, @⌘i. Assuming that @� acts as a di↵erentialoperator on g(@↵, @�) = g↵� according to

@� [g(@↵, @�)] = h@� [@↵] , @�i+ h@↵, @� [@�]i (61)

which we write as

g↵�,� = ha�↵, @�i+ h@↵, a��i

= g�⌘n

⌘�↵

o

+ g↵⌘n

⌘��

o

(62)

With historical hindsight, the combination of functions

g↵�,� + g��,↵ � g�↵,� (63)

can be solved forn

⌘↵�

o

asn

⌘↵�

o

=1

2g�⌘{↵�,�}. (64)

where15

{↵�,�}�= �g↵�,� + g�↵,� + g��,↵. (65)

The combination of partial derivatives of the metric coe�cients denoted {↵�,�} and given by eq. (65) arecalled the Christoffel symbols of the first kind while the raised form denoted

n

⌘↵�

o

and given by

eq. (64) are called the Christoffel symbols of the second kind16.We emphasize that finding the solution to eq. (63) hinges on the fact that, as follows from eq. (60),

g�⌘n

⌘↵�

o

is symmetric in lower ↵� indices.

15Like Schouten, we use the curly brackets {ijk} around a triple of indicies ijk to denote the cyclic sum S(i, j, k) =

ijk + jki+ kij with minus sign on the ”seed” ordering. That is, {ijk}�= �

ijk

+ jki

+ kij

16Schouten reference to Christo↵el 1869

20

Now that we have formulas for b↵� andn

⌘↵�

o

we revisit eq. (58) which states that the tangential

component of the rate of change of the velocity @↵ in the @� is given by

@� [@↵]� b↵� n = @⌘n

⌘↵�

o

. (66)

where the Christo↵el symbols encode (for each @↵ and @�) the information necessary to write @� [@↵] as alinear combination of the basis vectors @⌘.

6.2.2 Transformation Law of the Christo↵el Symbols: Importance of Coordinated Frames

Consider i and i0 and ↵ and ↵0 for transformaed frames ToDo

An important formula can be discovered by asking the following question.

Q: How does the formula for the Christo↵el symbols change if one selects a di↵erent pair of basisvectors vi (i = 1 . . . 2) for TS?

This question is answered by direct computation using vi = M↵i @↵ =) M i↵vi = @↵ with M↵i = M↵i (u�) aninvertible change of basis matrix (of functions of the coordinates u↵) with inverse M i↵. The key computation,which uses the bilinearity of g(·, ·) and vk behaves as a di↵erential operator on functions17 , is as follows

vk[gij ] = vk[g(vi, vj)]

= vk[g(M↵i @↵,M

�j @�)]

= vk[M↵i M

�j ]g↵� +M

↵i M

�j M

�k @�[g↵� ]

= g↵�M↵j

⇣

2v(kM�i)

⌘

+M↵��ijk g↵�,�| {z }

�

=F1

(↵��:ijk)

, (67)

where F1(↵�� : ijk) is viewed as a function of the “old” coordinate indices ↵,�,� and the “new” vectorindices i, j, k.

Notational Note: (On the Symmetric and Skew-Symmetric Bracket) The round brackets aresymmetric brackets and the square brackets are skew-symmetric brackets defined by

2v(jM↵i)

�= vj [M

↵i ] + vi[M

↵j ],

2v[jM↵i]

�= vj [M

↵i ]� vi[M↵j ].

Using eq. (67) three times to compute

�vk[gij ] + vi[gjk] + vj [gki] (68)

with the specified permuation of indices

�F1(↵�� : ijk) + F1(��↵ : jki) + F1(�↵� : kij)

and simplifying using the symmetry g↵� = g�↵ yields

� vk[gij ] + vi[gjk] + vj [gki]

= {↵�,�}M↵��ijk + g�↵M�k

⇣

2v(jM↵i)

⌘

+ g↵�M�j

⇣

2v[iM↵k]

⌘

+ g↵�M↵i

⇣

2v[jM�k]

⌘

. (69)

17By way of computation, when we write, for example, vk

[M↵i

] we mean M�k

@�

⇥M↵

i

⇤= M�

k

M↵i,�

where M↵i,�

makes sense

as partial di↵erentiation of the ↵th row and ith column of M denoted M↵i

w.r.t. u� because each entry of M is a function ofthe coordinates u� .

21

Without further assumptions on M and v[M ], eq. (69) is as far as we can go simplification wise. Since eq.(64) is given relative to a coordinate system u↵ and coordianted vectors @↵ it stands to reason that eq. (69)might be more useful when vk are themselves coordinated vectors (relative to new set of coordinates of thesurface S). By the Frobenius Theorem (of vector fields), such a coordinate system, say wi, can be found

if and only if the commutator bracket of vi and vj defined and denoted by [vi, vj ]�= vivj � vjvi is zero18.

In this coordinate system wi, the vectors vi are actually @i and hence M↵i = �↵,i where the old coordinates

u↵ are expressed in the new coordinates wi in terms of the coordinate functions u↵ = �↵(wi). Under thiscoordinated vector assumption, a new symmetry condition presents itself, for example,

vj [M↵i ] = @j [�

↵,i] = �

↵,ij =) �↵,ij = �↵,ji (70)

and eq. (69) simplifies further to

{ij,k} = 2g�↵��,k�↵,ij + {↵�,�}�↵,i��,j�

�,k (71)

Applying 12gkn = 12�

k,⌘�

n,�g

⌘� to both sides of eq. (71) yields

�

nij

= �n,⌘�⌘,ij +

�

�↵�

�↵,i��,j�

n,�, (72)

which is the transformation law (or change of coordiantes formula) of the Christo↵el symbols. Equation (72)is a foundational equation that, no matter the introduction of new ideas or notation, we can appeal to forgrounding.

For example, while the derivation of the Christo↵el symbols and its associated transform law was derivedand explained in a very geometric fashion, one would like to not have to redo this process every time if thedetails of the formal expression eq. (72) are forgotten19. In an e↵ort to formalize this equation we introducethe following notation.

Notational Note: (On Notation For Tangential Component of Acceleration) We use the notationr@� [@↵] for the tangential component of the acceleration @� [@↵]. Consequently, the notational definition ismade

r↵@��=n

⌘↵�

o

@⌘. (73)

We emphasize that r↵[@� ] is only a short-hand notation at this moment for the formula @⌘n

⌘↵�

o

where the

Christo↵el symbolsn

⌘↵�

o

are defined by eq. (64).

6.2.3 Christo↵el Connectionn

ijk

o

Defines Covariant Di↵erentiation rj@k =n

ijk

o

@i:

Leibnitzian Properties of Covariant Di↵erentiation

Given its geometric origins (as a partial derivative followed by a projection onto the tangent plane), thenotation r� is meant to evoke the feeling that a new kind of derivative (in the @� direction) has beenintroduced. What properties should this derivative have? Like the partial derivative @� , which acts as alinear operator on functions and satisfies the product (Leibniz) rule,

@� [M↵i @↵] = @� [M

↵i ]@↵ +M

↵i @� [@↵]

or

@� [M↵i @↵] = M

↵i,� +M

↵i @↵,�

18For a full proof see Frankel pgl 243. For a sketch of a proof let ✓i be the dual 1–forms of vi

6= 0 and hence ✓j(vi

) = �ji

.By Cartan’s formula, d✓j(v

i

, vk

) = vk

[✓j(vi

)] � vi

[✓j(vk

)] � ✓j([vi

, vk

]) = �✓j([vi

, vk

]. When [vi

, vk

] = 0, then d✓j = 0 whichsays that each of the 1–forms ✓j is exact and so, by the converse to Poincare’s lemma, ✓j = dwj where wj are locally definedfunctions (coordinates). It follows now that ✓j(v

i

) = dwj(vi

) = �ji

=) vi

= @i

and consequently that M↵i

= �↵,i

where

u↵ = �↵(wi) are the coordinate functions of u↵ relative to the new coordinats wi.19Of course, the idea behind the equation can not be forgotten since we have expossed it to the light of geometric intuition,

but it is not unexpected that the actual equation itself might be forgotten.

22

we would like r� to extend as a notation to linear combinations by the following assumed (symbolic)Leibnitz properties

(L1) r� [M↵i @↵] = r� [M↵i ]@↵ +M↵i r� [@↵]= M↵i,�@↵ +M

↵� @↵;� (74)

(L2) rM�i @� [@↵] = M�i r� [@↵]. (75)

The implications of L1,L2 are best shown in the computation

�

ijk

@i�= rj [@k]

= r�↵,j@↵ [��,k@� ]

= �↵,j

⇣

��,k↵@� + ��,kr↵[@� ]

⌘

= �↵,j

⇣

��,k↵ + ��,k

�

�↵�

⌘

@�

=⇣

��,jk + �↵,j�

�,k

�

�↵�

⌘

@�

= ��,jk@� + �↵,j�

�,k@�;↵

=⇣

�i,���,jk + �

i,��

↵,j�

�,k

�

�↵�

⌘

@i

=)�

ijk

= �i,���,jk + �

i,��

↵,j�

�,k

�

�↵�

(76)

which is the transformation law for the Christo↵el symbols. The point of this computation being that thenotational extension of r↵ was appropriate since, without it, the correct transformation law of the Christof-fel symbols would not have been obtained. Also the r↵ notation and its assumed properties made it easyto derive the transformation law of the Christo↵el symbols.The Christoffel (Levi-Civita) connection on a surface S ⇢ R3 is, for each coordinate system u↵,an assignment of the n(n

2+n)2 functions (Christo↵el symbols)

n

�↵�

o

. Furthermore, the relationship between

the functionsn

�↵�

o

andn

ijk

o

in two di↵erent coordinate systems u↵ and ui is given by eq. (72). 20

From now on we call r� = r@� the covariant derivative (in the @� direction) and r� [@↵] the covariantderivative of @↵ (in the @� direction). By our notational definition r� [@↵] = @�

n

��↵

o

wheren

��↵

o

are the

Christo↵el symbols we can be a little more descriptive and say that r� [@↵] = @�n

��↵

o

is the covariant

derivative of @↵ with respect to the Christoffel (Levi-Civita) connection.

To this point we have only defined the covariant derivative of the vectors @↵ in the directions @� . Using bothgeometric and symbolic properties of covariant di↵erentiation, we now extend the covariant derivative to avector Y = Y ↵@↵. Viewing @� again as a di↵erential operator leads to

@� [Y ] = @� [Y↵]@↵ + Y

↵@� [@↵]

= Y ↵,�@↵ + Y↵a�↵

eq.(58)= Y ↵,�@↵ + Y

↵�

@��

��↵

+ b�↵, n�

=�

Y �,� +�

��↵

Y ↵�

@�| {z }

�

=r� [Y ]

+b�↵Y↵ n. (77)

Consistent with earlier notation, denote the tangential component of @� [Y ] as r� [Y ] with formal definitiongiven by

r� [Y ] = Y �;�@� with Y �;��= Y �,� +

�

��↵

Y ↵. (78)

In other words, for each direction @� , r@� [Y ] is a tangent vector with component functions obtained byfirst computing the @� directional derivative of the original component functions Y � of Y , Y �,� and then

20[Spivak,pg. 240] points out that this connection should be called the Christo↵el connection, but due to a historical mixup,it is more often goes by the name Levi-Civita conenction. Also, following Spivak’s terminology, the Christo↵el connection asdefined here is called a classical connection.

23

compensating by the term involving the Christo↵el symbolsn

��↵

o

Y ↵. Using the Leibnitizian properties

L1 and L2, we show now that, unlike the components Y �,� of the directional derivative @� [Y ] = Y↵,�@↵, the

components Y �;� of the covariant derivative r� [Y ] do transform as a tensor. Observe

Y k;j@k = rj [Y i@i] = ��,jr� [Y

i�↵,i�j,↵@j ]

= ��,jr� [Y↵@↵]

= ��,j�

Y ↵,� +�

��↵

Y ↵�

@�

= ��,j�k,�Y

�;�@k

=) Y k;j = ��,j�

k,�Y

�;�

Having extended r� [@↵] to r� [Y ], it remains to discuss rX [Y ] where X = X�@� . Geometrically, rX [Y ] isthe tangential component of X[Y ] and, following eq. (77), is given by the formula

rX [Y ] = X�Y �;�@�. (79)

We now restrict our attention to extending the covariant derivative from vectors Y to covectors ✓ andultimately to the metric tensor g(·, ·) = h·, ·i. Geometrically, our intuition for extending the covariantderivative is a little lacking, and rather relies more on properties we believe the the covariant derivativeshould possess. For example, the covariant derivative r� of the coordinated covectors du↵ should again bea combination of the covectors du� given by r� [du↵] = M↵��du� for some M↵��. How should we determineM? As addressed earlier, the quantity ! = du↵ ⌦ @↵ is the identity transformation on vectors X. It makessome intuitive sense that as a kind of derivative the covariant derivative of a constant (namely, the diagonalentries of !) then r�! = 0. Assuming that r� distributes to each component of the tensor product

0 = r� [!] = r� [du↵ ⌦ @↵]= r� [du↵]⌦ @↵ + du↵ ⌦r� [@↵]= M↵��du

� ⌦ @↵ +�

�↵�} du↵ ⌦ @�

=�

M��↵ +�

��↵}

�

du↵ ⌦ @�=) M��↵ = �

�

��↵

.

It follows thatr�du� = �

�

��↵

du↵. (80)

Following the same process as that leading to eq. (78) and eq. (79), the covariant derivative r� and rX ofthe covector ✓ = ✓�du� is

r�✓ = ✓↵;�du↵ (81)rX✓ = X�✓↵;�du↵ (82)

where ✓↵;��= ✓↵,� �

n

��↵

o

✓�. With the covariant derivative rX of vectors X and covectors ✓, extensionto higher tensors is straightforward. For example, the covariant derivative of the metric tensor or firstfundamental form

g = g↵�du↵ ⌦ du� ,

denoted rXg is itself again a (0,2) tensor, and therefore acts on two vectors Y, Z to give

(rXg) (Y, Z) =rX [g(Y, Z)] + g(rX [Y ], Z) + g(Y,rX [Z])=) rXg = X�g↵�;�du↵ ⌦ du� (83)

where g↵�;� = g↵�,� � g↵⇠n

⇠��

o

� g⇠��

⇠↵�

. Using the definition of the Christo↵el symbols and that

�↵g�,� = (�↵g�),� � �↵,�g� = g↵�,� , it is a straightforward substitution and computation which shows

24

that cancellation happens in pairs and therefore g↵�;� = 0,

g↵�;� = g↵�,� � g↵⇠n

⇠��

o

� g⇠��

⇠↵�

= g↵�,� � g↵⇠✓

1

2g⇠(�g��, + g�,� + g�,�)

◆

� g⇠�✓

1

2g⇠(�g↵�, + g↵,� + g�,↵)

◆

= g↵�,� �1

2�↵ (�g��, + g�,� + g�,�)

� 12�� (�g↵�, + g↵,� + g�,↵)

= g↵�,� � g↵�,� +1

2g��,↵ �

1

2g��,↵

+1

2g↵�,� �

1

2g↵�,� = 0.

Since g↵�;� = 0 it follows that rXg = 0 for all X and we say that the metric tensor is covariant constant (orthat the covariant derivative is metric comptaible). It follows now that the covariant derivative w.r.t. theChristo↵el connection distributes across the metric tensor21 as

rX [g(Y, Z)] = �g(rX [Y ], Z)� g(Y,rX [Z]). (84)

To set ourselves up for future extensions of the covariant derivative define, with hindsight and no geometricmotivation22, the torsion vector field T (X,Y ) or torsion tensor T = T �↵�du

↵ ⌦ du� ⌦ @� as the di↵erencebetween the commutator bracket and the covariant derivatives

T (X,Y )�= [X,Y ]� (rX [Y ]�rY [X]) . (85)

The implications of this equation follow from its direct computation

T (@↵, @�) = (@↵[@� ]� @� [@↵])� (r↵[@� ]�r� [@↵])= (b↵� � b�↵) n= 0 since b↵� = b�↵.

Alternatively, using the fact that [@↵, @� ] = 0

T (@↵, @�) = (@↵[@� ]� @� [@↵])� (r↵[@� ]�r� [@↵])=�

��

�↵�

+�

��↵

�

@�

= 0 since by definition�

�↵�

=�

��↵

.

For general vector fields, X = X↵@↵ and Y = Y �@� the Leibniz property of @� and r� leads to

[X,Y ] =�

X↵Y �,↵ � Y �X�,��

@�

rX [Y ]�rY [X] =�

X↵Y �,↵ � Y �X�,��

@�

+X↵Y �T (@↵@�)

and therefore the torsion vector

T (X,Y ) = X↵Y �T (@↵@�)

= X↵Y ��

��

�↵�

+�

��↵

�

@� (86)

is zero23.21While eq. (84) relates r

X

and the metric tensor, and therefore does contain some kind of geometric information, it is notimmediately clear to this author at this point exactly what this information is. The geometric usefulness of this equation willmake itself clear when we look at connections and covaraint derivatives from the moving frame perspective.

22Though we could talk about the flow along the vector field and parallel translation along vector field with torsion thequantity which measures the di↵erence.

23Whatever that means geometrically. Still working on this point. See Schouten pg. 126 and 127 for Cartan’s explanationthrough Schouten.

25

6.3 The Shape Operator, b↵�In the previous section we defined the Christo↵el symbols in terms of the tangential component of the

“acceleration” a↵� , ha↵� , @⌘i�= g�⌘

n

�↵�

o

24 In this section, we focus on the normal component of acceleration

b↵��= ha↵� , ni. Since @↵ and n are perpindicular, h@↵, ni = 0, it follows that

@� [h@↵, ni] = ha↵� , ni+ h@↵, @�ni = 0=) b↵� = h@↵,�@�ni. (87)

That is, b↵� is straight–forward to compute as

�ijni�j,↵� = nj�

j,↵� = (n ·D

2�)↵� or

�ijni,��

j,↵ = nj,��

j,↵ = (Dn)j� · (D�)j↵ = (Dn ·D�)↵�

which is the ↵–the row and �–th column entry of the Jacobian of the normal vector matrix–matrix timesthe Jacobian of the coordinate transform function. Consider the term �@�n, which as a vector in R3, canbe written as �@�n = ni,�@i for i = 1 . . . 3. This vector is actually in the tangent space because n is a unitvector hn, ni = 1 which implies

@� [hn, ni] = 0 =) h@�n, ni = 0. (88)

Therefore we can re-write the vector �@�n as some yet undetermined linear combination of the tangentvectors @↵ according to

�@�n = ni,�ei = b��@� (undetermined, yet strategically chosen notation b��), (89)

and further, since both input vectors are in the tangent plane, we can write eq. (87) as

b↵� = ha↵� , ni = g(@↵,�@�n) = g↵�b�� .

It follows that the undetermined matrix of functions b�� is found by

b�� = g↵�b↵� . (90)

With the functions b�� determined and recognizing that, as discussed in the previous section, r�=tangentialcomponent of @� the Weingarten Equations follow as

�r�n = b��@�, (91)

and can be expressed as an operator (or vector–valued one form) called the Shape Operator25

�rn = b��du� ⌦ @�, (92)

mapping tangent vectors X = X�@� to a surface to tangent vectors according to

X �rn = �rXn = b��X�@�, (93)

and more specifically @� rn = b��@�. The shape operator is linear (a) (X+Y ) rn = X rn+Y rnand (b) fX rn = f ·(X rn) by properties of the covariant derivative. One might righfully call both b��and b↵� the shape operator, but we reserve this term for b↵� and think of b↵� as being metrically equivalent to

(or the lowered form of) the shape operator b↵� .26 We can further express the functions b↵� = g(�r�n, @↵)

24Recall footnote10 where we discussed the notational convections surrounding h·, ·i and g(·, ·). Namely, we use h·, ·i whenat least one input vector “points” out of the tangent plane and we use g(·, ·) when both vectors are in the tangent plane to asurface.

25Again, in a slight abuse of notation we do not distinguish at this moment between the Shape Operator �rn = b��

du�⌦@�

and the components of the shape operator b��

.26This fact is corroborated by Oneill, pg. 190 who also defines the shape operator in terms of �r

v

U where v is a vector inthe tangent space and U is a unit normal vector.

26

as a (0,2)–tensor by making it a “ function of the direction vectors” to obtain the Second FundamentalForm

b = b↵�du� ⌦ du↵ = g(�r·n, ⇤)

= g↵�b��du

� ⌦ du↵

= b↵�du� [·]⌦ du↵[⇤].

The action of the second fundamental form on two tangent vectors, X and Y is given by

b(X,Y ) = g(�r·n, ⇤)(X,Y ) = b↵�du� [X]⌦ du↵[Y ]= b↵�X

�Y ↵,

and more specifically that b(@↵, @�) = b↵� . It is not unexpected that there could be some concern aboutwhich index goes where in the second fundamental form and therefore that we need to be worried aboutwhere the lowered index goes in the formula

b↵� = g(@↵,�r�n) = g↵�b�� . (94)

That is, which index in b↵� is the directional derivative index � and which is the vector to compute the angletowards ↵? It turns out that the second fundamental form is symmetric and therefore b↵� = b�↵, whichalleviates our concern about which index goes where. That this form b↵� is symmetric follows easily fromthe fact that @↵[@� ] = @� [@↵] since @� [@↵] = @� [@i↵ei] = @� [�

i,↵ei] = �

i,↵�ei where the derivative ,� of the

Jacobian �i,↵ is symmetric. Now that we have shown the components b↵� of the second fundamental formare symmetric b↵� = b�↵ it follows that the shape operator is self adjoint (w.r.t the standard inner (dot)product) since

b↵� = g(�rn[@↵], @�) = g(@↵,�rn[@� ]) = b�↵ (95)

and therefore the shape operators has real eigenvalues and the eigenvectors corresponding to the distincteigenvalues are orthogonal.27 For notational bookeeping let’s define the mapping (to obtain the normalcomponent of acceleration) as

?

rX�= b(X, ·)n = b↵�X�du↵ ⌦ n, (96)

and more specifically?

r�= b(@� , ·)n = b↵�du↵ ⌦n. That is, the mapping?

rX takes tangent vectors as inputand returns a normal vector. In this way, as previously discussed, we may write the accelerations a↵� intangential and normal components

a↵� = @� [@↵]

= (r�+?

r�)[@↵]

= r�@↵+?

r� @↵=�

��↵

@� + b�↵n.

From prior properties of the covariant derivation r and the tensorial nature of the metric tensor g(·, ·) (whichendows

?

r with linearity properties also), it follows that

X[@↵] = rX [@↵]+?

rX [@↵]=�

��↵

X�@� + b�↵X� n

X[Y ] = rXY+?

rX Y= X�Y �;�@� + b�↵X

�Y ↵ n.

27When the eigenvalues of the shape operator are distinct, it is a small step now to look at the curves on the surface inthese principle or eigendirections and to then discuss the curve curvature of said principal curves. In this way we can build afeeling for the Gauss Curvature of the surface as the product of these principal curvatures. Further, the matrix of the shapeoperator, relative to the principal basis is diagonalized (with principal curvatures on the diagonals) and hence the determinantof the shape operator is also the Gauss Curvature. See Oneill, pg. 203.

27

Extending the “first derivative operator” @↵ = r↵+?

r↵ to the “second derivative operator” @2�� yields

@2��[@↵] = @� [@�@�] = @�h

r�@↵+?

r� @↵i

= r� [r�@↵] +?

r� [r�@↵] +r�h?

r� @↵i

+?

r�h?

r� @↵i

,

which can be grouped into tangential and normal components as

@2��[@↵] = r� [r�@↵] +r�h?

r� @↵i

+?

r� [r�@↵] +

=0z }| {

?

r�h?

r� @↵i

=

tangentialz }| {

r2�� [@↵] +

tangential/normalz }| {

r�h?

r� @↵i

+

normalz }| {

?

r� [r�@↵] .

Expanding the non–zero terms yields

@2��[@↵] = r�hn

⇢�↵

o

@⇢i

+r� [b�↵n] +?

r�hn

⇢�↵

o

@⇢i

=

✓

n

⌘�↵

o

,�+n

⇢�↵

o

�

⌘�⇢

◆

@⌘| {z }

tangential

+ b�↵,� n+ b�↵b⌘�@⌘

| {z }

tangential/normal

+n

⇢�↵

o

b�⇢ n| {z }

normal

where the the normal term is computed via the definition of the second fundamental form by

?

r� [r�@↵]�= b

⇣

@�,n

⇢�↵

o

@⇢⌘

n = g⇣

�r�n,n

⇢�↵

o

@⇢⌘

n =n

⇢�↵

o

b⇠�g⇠⇢ n = b�⇢n

⇢�↵

o

n.

Alternatively, the same computation for the “second derivative operator” can be computed (perhaps moresimply) as

@2��[@↵] = @�hn

⇢�↵

o

@⇢ + b�↵ni

=n

⇢�↵

o

,�@⇢ +

n

⇢�↵

o

@�[@⇢] + b�↵,� n� b�↵@�n

=

✓

n

⌘�↵

o

,�+�

�↵

{⌘�}� b↵�b⌘�◆

@⌘ +⇣n

⇢�↵

o

b⇢� + b↵�,�⌘

n

=⇣

R···⌘��↵· � b↵�b⌘�

⌘

@⌘| {z }

tangential

+⇣

b⇢�n

⇢�↵

o

+ b↵�,�⌘

n| {z }

normal

where we are, for convenience, defining the (pre-curvature) symbols

R···⌘��↵·�=n

⌘�↵

o

,�+�

�↵

{⌘�} . (97)

It follows then that skew-symmetrizing on the � and � derivative indices

2@2[��] [@↵] =⇣

R···⌘��↵· � 2b↵[�b⌘�]

⌘

@⌘

+ 2⇣

b⇢[�n

⇢�]↵

o

+ b↵[�,�]⌘

n, (98)

where the intrinsic curvature tensor R···⌘��↵· is defined as

R···⌘��↵·�= 2R···⌘[��]↵· (99)

28

Using the fact that mixed partials commute (and therefore 2@2[��] [@↵] = 0) yields the integrability con-

ditions28 of Gauss29 and Codazzi and Peterson30, respectively,

R···⌘��↵· = 2b↵[�b⌘�] (100)

b⇢[�n

⇢�]↵

o

+ b↵[�,�] = 0 (101)

As addressed in an earlier footnote, we know that the Gauss Curvature K of a surface equals thedeterminant of the shape operator. We also know the relationship of the shape operator to the secondfundamental form and therefore 31

K = det(b↵� ) = det(g↵�b��) =

det(b��)

det(g↵�). (102)

And while it seems that the Gauss curvature K of a surface requires knowledge of the embedding space R3the integrability equation of Gauss (of which, due to the skew–symmetry in the derivative indices � and �,only the R···⌘12↵· entry survives), shows that, in lowered form,

R12↵⌘ = 2b↵[1b2]⌘ = b↵1b2⌘ � b↵2b1⌘. (103)

However, due to the symmetry of the second fundamental form, the diagonal entries (↵, ⇠) = (i, i), i = 1 . . . 2of R12↵⌘ are zero because, for example,

R1212 = b11b21 � b12b11 = 0 since b21 = b12. (104)

In fact, as a matrix, R12↵⌘ is skew–symmetric and therefore R12↵⌘ = �R12⌘↵ and therefore the only termsto survive of R12↵⌘ is

R1212 = b11b22 � b12b21 = det(b↵�)!! (105)

Therefore, the Gauss curvature K simplifies into completely intrinsic terms (i.e. terms involving only themetric tensor and its first and second derivatives) and we have Gauss’ Theorema Egregium (RemarkableTheorem)

K =R1212

det(g↵�). (106)

28Recall that the metric coe�cients g↵�

and the shape operator b↵�

can be used to defined the first and second fundamentalforms g = g

↵�

du↵ ⌦ du� and b = b↵�

du↵ ⌦ du� . These integrability conditions are those equations that must be necessarilysatisfied if the matrices g

↵�

and b↵�

are to be the fundamental forms of a given surface. It is beyond the scope of these notes,but evidently Bonnet showed that these are also su�cient to ensure the local existence of a surface in R3 with prescribed g

↵�

and b↵�

.29We will have much more to say about the symbols R···⌘

��↵· in later sections. Su�ce it to say at the moment that these

symobls are the intrinsic curvature tensor (of the Levi-Civita/Christo↵el symbols).30At the moment I don’t see doing much with the Codazzi and Peterson equation.31The latter part of this equation compares favorably with the curve curvature case where one = aN

v·v =normal componentof acceleration/speed.

29

6.4 Example Computations: First and Second Fundamental Forms g↵� and b↵�, Christo↵el Symbols, Shape Operator, and Gauss Curvature, K

6.4.1 Geometric Data for Surface of Sphere in R3 Described by “Angular Coordinates”

(1)(1)

"Coordinated Frame on Sphere Computations (r constant):"

Fi =

r cos q sin f

r sin q sin f

r cos f

, F, ai

=

Kr sin q sin f r cos q cos f

r cos q sin f r sin q cos f

0 Kr sin f

F, a bi

=

Kr cos q sin f Kr sin q cos f

Kr sin q cos f Kr cos q sin f

Kr sin q sin f r cos q cos f

r cos q cos f Kr sin q sin f

0 0

0 Kr cos f

, ni =

sin f cos q

sin f sin q

cos f

"Sphere_Vectors" =Dq

= r cos q sin f DyK r sin q sin f D

x

Df

= Kr sin f DzC r sin q cos f D

yC r cos q cos f D

x

ga b

=r2 sin f 2 0

0 r2, ga b =

1

r2 sin f 20

01r2

Ga bs

=

0, Ksin f cos f cos fsin f

, 0

cos fsin f

, 0 0, 0

, Ga bs

Ds

=

Ksin f cos f Df

cos f Dq

sin f

cos f Dq

sin f0

Ra b s d

=

0 00 0

0 r2 sin f 2

Kr2 sin f 2 0

0 Kr2 sin f 2

r2 sin f 2 0

0 00 0

ba b

=Kr sin f 2 0

0 Kr, b

ab =

K1r 0

0 K1r

,R1212 = Det ba b

R1212 = r2 sin f 2

,

K =R1212

Det ga b

K =1r2

Figure 5: Sphere Frame Data (computations done in Maple using FramesDataComps.mw). Computa-tions done according to: (a) �i=change of variables from Surface coordinates to Cartesian coordinates,(b) �i,↵=Jacobian of �

i, (c) �i,↵�=vector of Hessian matrices of functions �i, (d) g↵� = g(@↵, @�) =

h@↵, @�i=First Fundamental Form= matrix of dot product of surface vectors, (e) g↵�=inverse of g↵� ,(f) ��↵� =

n

�↵�

o

=Christoffel symbols (of second kind)=specific combination of first derivatives of g↵�

according to eq. (64), (g) b↵� = h�i↵� , nii=(components of) Second Fundamental Form=dot productof vector of Hessians with normal vector to surface, (h) b�↵=Shape Operator, (i) R1212=only non-zerocomponent (up to symmetries) of Riemann (Intrinsic) Curvature Tensor computed by combinationof first derivatives of ��↵� according to eq. (99), (j) K=Gauss Curvature

30

6.4.2 Geometric Data for Surface of Cylinder in R3

(3)(3)

"Coordinated Frame on Cylinder Computations (r constant):"

Fi =

r cos q

r sin qz

, F, ai

=

Kr sin q 0

r cos q 00 1

F, a bi

=

Kr cos q 00 0

Kr sin q 00 0

0 00 0

, ni =

cos q

sin q

0

"Cylinder_Vectors" =Dq

= r cos q DyK r sin q D

x

Dz = Dz

ga b

=r2 00 1

, ga b =

1r2

0

0 1

Ga bs

=0, 0 0, 00, 0 0, 0

, Ga bs D

s=

0 00 0

, Ra b s d

=

0 00 0

0 00 0

0 00 0

0 00 0

ba b

=Kr 00 0

, bab =

K1r 0

0 0,R

1212= Det b

a b

R1212

= 0,K =

R1212

Det ga b

K = 0

Figure 6: Cylinder Frame Data (computations done in Maple using FramesDataComps.mw). Computa-tions done according to: (a) �i=change of variables from Surface coordinates to Cartesian coordinates,(b) �i,↵=Jacobian of �

i, (c) �i,↵�=vector of Hessian matrices of functions �i, (d) g↵� = g(@↵, @�) =

h@↵, @�i=First Fundamental Form= matrix of dot product of surface vectors, (e) g↵�=inverse of g↵� ,(f) ��↵� =

n

�↵�

o

=Christoffel symbols (of second kind)=specific combination of first derivatives of g↵�

according to eq. (64), (g) b↵� = h�i↵� , nii=(components of) Second Fundamental Form=dot productof vector of Hessians with normal vector to surface, (h) b�↵=Shape Operator, (i) R1212=only non-zerocomponent (up to symmetries) of Riemann (Intrinsic) Curvature Tensor computed by combinationof first derivatives of ��↵� according to eq. (99), (j) K=Gauss Curvature

31

6.4.3 Geometric Data for Surface z=f(x,y) in R3

(14)(14)

"Surface (x,y,f(x,y))"

Fi =x

y

f

, F, ai =

1 0

0 1

fx fy

, F, a bi =

0 0

0 0

0 0

0 0

fx, x fx, yfx, y fy, y

"Tangent_Vectors_To_Surface" =vx = fx DzC Dxvy = fy DzC Dy

ga b =1 C fx

2 fx fy

fx fy 1 C fy2

, ga b =

1 C fy2

fx2C fy

2C 1K

fx fyfx

2C fy2C 1

Kfx fy

fx2C fy

2C 1

1 C fx2

fx2C fy

2C 1

Ga bs

=

fx fx, xfx

2C fy2C 1

,fx, x fy

fx2C fy

2C 1

fx fx, yfx

2C fy2C 1

,fx, y fy

fx2C fy

2C 1

fx fx, yfx

2C fy2C 1

,fx, y fy

fx2C fy

2C 1

fx fy, yfx

2C fy2C 1

,fy fy, y

fx2C fy

2C 1

ba b =

fx, x

fx2C fy

2C 1

fx, y

fx2C fy

2C 1

fx, y

fx2C fy

2C 1

fy, y

fx2C fy

2C 1

bab =

Kfx, y fx fyK fx, x fy

2K fx, x

fx2C fy

2C 1 3/2fx, y fy

2K fy, y fx fyC fx, y

fx2C fy

2C 1 3/2

fx, y fx2K fx, x fx fyC fx, y

fx2C fy

2C 1 3/2Kfx, y fx fyK fy, y fx

2K fy, y

fx2C fy

2C 1 3/2

R1212 = Det ba b

R1212 = KKfx, x fy, yC fx, y

2

fx2C fy

2C 1

,

K =R1212

Det ga b

K = KKfx, x fy, yC fx, y

2

fx2C fy

2C 1 2

Figure 7: Frame Data (computations done in Maple using FramesDataComps.mw). Computations done ac-cording to: (a) �i=Parametrization of Surface, (b) �i,↵=Jacobian of �

i, (c) �i,↵�=vector of Hessian matrices

of functions �i, (d) g↵� = g(@↵, @�) = h@↵, @�i=First Fundamental Form= matrix of dot product ofsurface vectors, (e) g↵�=inverse of g↵� , (f) ��↵� =

n

�↵�

o

=Christoffel symbols (of second kind)=specific

combination of first derivatives of g↵� according to eq. (64), (g) b↵� = h�i↵� , nii=(components of) SecondFundamental Form=dot product of vector of Hessians with normal vector to surface, (h) b�↵=ShapeOperator, (i) R1212=only non-zero component (up to symmetries) of Riemann (Intrinsic) CurvatureTensor computed by combination of first derivatives of ��↵� according to eq. (99), (j) K=Gauss Curva-ture

32

6.4.4 Geometric Data for Top Half of Sphere in Cartesian Coordinates in R3

(16)(16)

"Surface (x,y,f(x,y))"

Fi =

x

y

r2K x2K y2, F, a

i =

1 0

0 1

Kx

r2K x2K y2K

y

r2K x2K y2

"Tangent_Vectors_To_Surface" =

vx = Kx Dz

r2K x2K y2C Dx

vy = Ky Dz

r2K x2K y2C Dy

ga b =

r2K y2

r2K x2K y2x y

r2K x2K y2

x yr2K x2K y2

r2K x2

r2K x2K y2

, ga b =

r2K x2

r2Kx yr2

Kx yr2

r2K y2

r2

Ga bs

=

x r2K y2

r2 r2K x2K y2, y r

2K y2

r2 r2K x2K y2x2 y

r2 r2K x2K y2, x y

2

r2 r2K x2K y2

x2 yr2 r2K x2K y2

, x y2

r2 r2K x2K y2x r2K x2

r2 r2K x2K y2, y r

2K x2

r2 r2K x2K y2

ba b =

Kr2K y2

r r2K x2K y2K

x yr r2K x2K y2

Kx y

r r2K x2K y2K

r2K x2

r r2K x2K y2

, bab =

K1r

0

0 K1r

Ra b s d =

0 0

0 0

0 1r2K x2K y2

K1

r2K x2K y20

0 K 1r2K x2K y2

1r2K x2K y2

0

0 0

0 0

R1212 = Det ba b

R1212 =1

r2K x2K y2,

K =R1212

Det ga b

K = 1r2

Figure 8: Frame Data (computations done in Maple using FramesDataComps.mw). Computations done ac-cording to: (a) �i=Parametrization of Surface, (b) �i,↵=Jacobian of �

i, (c) �i,↵�=vector of Hessian matrices

of functions �i, (d) g↵� = g(@↵, @�) = h@↵, @�i=First Fundamental Form= matrix of dot product ofsurface vectors, (e) g↵�=inverse of g↵� , (f) ��↵� =

n

�↵�

o

=Christoffel symbols (of second kind)=specific

combination of first derivatives of g↵� according to eq. (64), (g) b↵� = h�i↵� , nii=(components of) SecondFundamental Form=dot product of vector of Hessians with normal vector to surface, (h) b�↵=ShapeOperator, (i) R1212=only non-zero component (up to symmetries) of Riemann (Intrinsic) CurvatureTensor computed by combination of first derivatives of ��↵� according to eq. (99), (j) K=Gauss Curva-ture

33