2. Differentiable Manifolds And Tensors

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2. Differentiable Manifolds And Tensors
2.1 Definition Of A Manifold 2.2 The Sphere As A
Manifold 2.3 Other Examples Of Manifolds
2.4 Global Considerations 2.5 Curves
2.6 Functions On M 2.7 Vectors And Vector Fields
2.8 Basis Vectors And Basis Vector Fields
2.9 Fiber Bundles 2.10 Examples Of Fiber
Bundles 2.11 A Deeper Look At Fiber Bundles
2.12 Vector Fields And Integral Curves
2.13 Exponentiation Of The Operator
d/d? 2.14 Lie Brackets And Noncoordinate Bases
2.15 When Is A Basis A Coordinate Basis?
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2.16 One-forms 2.17 Examples Of One-forms
2.18 The Dirac Delta Function 2.19 The Gradient
And The Pictorial Representation Of A One-form
2.20 Basis One-forms And Components Of One-forms
2.21 Index Notation 2.22 Tensors And Tensor
Fields 2.23 Examples Of Tensors 2.24 Components
Of Tensors And The Outer Product
2.25 Contraction 2.26 Basis Transformations
2.27 Tensor Operations On Components
2.28 Functions And Scalars 2.29 The Metric
Tensor On A Vector Space 2.30 The Metric Tensor
Field On A Manifold 2.31 Special Relativity
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2.1. Definition of a Manifold
Rn ?? Set of all n-tuples of real numbers
A ( topological ) manifold is a continuous space
that looks like Rn locally.
Definition ( Topological ) manifold A manifold
is a set M in which every point P has an open
neighborhood U that is related to some open set
fU(U) of Rn by a continuous 1-1 onto map fU.
Caution Length is not yet defined.
xUj(P) are the coordinates of P under fU.
Dim(M) n.
M is covered by patches of Us.
Subscript U will often be omitted
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Let
be 1-1.
(U, f ) Chart
U,V open ?
open
Let
( Coordinate transformation )
Charts (U, f ) (V, g ) are Ck - related if
exist are continuous ? i, j
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Atlas Union of all charts that cover M M is a
Ck manifold if it is covered by Ck charts. A
differentiable manifold is a Ck manifold with k gt
1.

• Some allowable structures on a differentiable
  manifold
• Tensors
• Differential forms
• Lie derivatives

M is smooth if it is C?. M is analytic (C?) if
all coordinate transformations in it are analytic
functions.
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2.2. The Sphere As A Manifold
Every P has a neighborhood that maps 1-1 onto an
open disc in R2.
( Lengths angles not preserved )
Spherical coordinates

• Breakdowns of f
• North south poles in S2 mapped to line (0,
  x2) (?, x2) in R2, resp.
• Points (x1, 0) (x1, 2?) correspond to same
  point in S2.

?
2nd map needed
? Both poles semi-circle joining them ( ? 0
) not covered
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One choice of 2nd map Another spherical
coordinates with its ? 0 semi-circle given by
(?/2 , ? ) ? ? ?/2 , 3?/2 in terms of
coordinates of 1st system.
Stereographic map (fails at North pole only)
North pole mapped to all of infinity
These conclusions apply to any surface that is
topologically equivalent to S2.
The 2-D annulus bounded by 2 concentric circles
in R2 can be covered by a single ( not
differentiable everywhere ) coordinate patch
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2.3. Other Examples Of Manifolds
Loosely, any set M that can be parameterized
continuously by n parameters is an n-D manifold

• Examples
• Set of all rotations R(??,?,?) in E3. ( Lie
  group SO(3) )
• Set of all boost Lorentz transformations (3-D).
• Phase space of N particles ( 6N-D).
• Solution space of any ( algebraic or
  differential ) equation
• Any n-D vector space is isomorphic to Rn.
• A Lie group is a C?-manifold that is also a
  group.

Rn is a Lie group wrt addition.
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2.4. Global Considerations
Any n-D manifolds of the same differentiability
class are locally indistinguishable.
Let f M ? N If f f 1 are both 11 C?,
then f is a diffeomorphism of M onto N. M N are
diffeomorphic.

• Example of diffeomorphic manifolds
• Smooth crayon sphere
• Tea cup torus

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2.5. Curves
Parametrized curve with parameter ?
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2.6. Functions on M
f is differentiable if f(x1,,xn) ? C1
Reminder
Any sufficiently differentiable set of equations
that is locally invertible ( with finite Jacobian
) is an acceptable coordinate transformation
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2.7. Vectors and Vector Fields
Let
be a curve C? through P on M.
be a differentiable function on M.
?
is a differentiable function on C?.
Chain rule
?
d xi infinitesimal displacement along C?
d xi ?? components of vector tangent to C?
?
vector tangent to C? with components
wrt basis
Tangent (vector) space TP(M) at P
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• Advantages of defining vector as d/d?
• No finite distance involved ? works on
  manifolds without metric.
• Coordinate free.
• Conforms with the geometric notion that a
  tangent is generated by some infinitesimal
  displacement along the curve.

• Finer points
• Each d/d? denotes an equivalent class of
  diistinct curves having the same tangent at P.
• Two vectors d/d? d/d? may denote the same
  tangent to the same curve under different
  parametrization.
• Only vectors at the same point on M can be
  added to produce another vector. ( Vectors at
  different points belong to different vector
  spaces )

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2.8. Basis Vectors And Basis Vector Fields

• Let M be an n-D manifold.
• ? P ? M, TP(M) is an n-D vector space.
• Any collection of n linearly independent
  vectors in TP(M) is a basis for TP(M) .
• A coordinate system x i in a neighborhood U
  of P defines a coordinate basis ?? / ? x i
  at all points in U.

Let
be an arbitrary (non-coordinate) basis for TP(M) .
For a vector field, V i V j are functions on
M. The vector field is differentiable if these
functions are.
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?? / ? x i is a basis ? it is a linearly
independent set. This is guaranteed if ?x i
are good coordinates at P.
Proof
Let ?y i be another set of good
coordinates at P.
?
is invertible
Inverse function theorem ?
i.e., vectors with components given by the
columns of J are linearly independent .
The j th column
represents the vector
QED
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2.9.a. Bundles
Ref Choquet, III.2
Let E B be 2 topological spaces with a
continuous surjective mapping ? E ? B Then (
E, B,?? ) is a bundle with base B.
E.g. Cartesian bundle
with
Cartesian product
Bundle is a generalization of the topological
products. E.g., A cylinder can be described as
the product of a circle S1 with a line segment
I. A Mobius strip can only be described as a
bundle.
If the topological spaces ?1(x) are
homeomorphic to a space F ? x?B, then ?1(x) is
called a fibre Fx at x, and F is a typical
fibre.
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2.9. Fibre Bundles

• A fibre bundle ( E, B, ?, G ) is a bundle ( E, B,
  ? ) with a typical fibre F,
• a structural group G of homeomorphism of F onto
  itself,
• and a covering B by a family of open sets Uj
  j?J such that
• The bundle is locally trivial ( homeomorphic to a
  product bundle )

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b) Let
then
is an element of G
c) The induced mappings
are continuous
These transition functions satisfy
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A vector bundle is a fibre bundle where the
typicle fibre F is a vector space the
structural group G is the linear group.
The tangent bundle
is a vector bundle with F Rn.
In a coordinate patch (U,x) of M, the natural
basis of Tp(U) is ?/ ?xi. The natural
coordinates of T(U) are ( x1, , xn, ?/ ?x1, ,
?/ ?xn ) ( x, v ).
A vector field is a cross section of a vector
bundle.
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2.10. Examples of Fiber Bundles

• Tangent bundle T(M).
• Tensor field on manifold.
• Particle with internal, e.g., isospin, state.
• Galilean spacetime B time, F E3.
• Time is a base since every point in space can be
  assigned the same time in Newtonian mechanics.
• Relativistic spacetime is a manifold, not a
  bundle.
• Frame bundle derived from T(M) by replacing
  Tp(M) with the set of all of its bases , which is
  homomorphic to some linear group. ( see
  Aldrovandi, 6.5 )
• Principal fibre bundle F ? G. E.g. frame
  bundle.

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2.11. A Deeper Look At Fiber Bundles
A fibre bundle is locally trivial but not
necessarily globally so. ? generalizes product
spaces
Example T(S2) Schutzs discussion is faulty. It
is true that S2 has no continuous cross-section ?
frame bundle of S2 is not trivial. For proof
see Choquet, p.193. However, the (Brouwer) fixed
point theorem states that every homeomorphism
from a closed n-ball onto itself has at least one
fixed point. It says nothing about n-spheres Sn.
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Example see Choquet, p.126. Mobius band Fibre
bundle ( M, S1, p, G ) with typical fibre F I ?
R. The base S1 is covered by 2 open sets. G
e, s ? C2 .
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2.12. Vector Fields And Integral Curves
Vector field A rule to select V(P) from TP ?
P ? M. Integral curve of a vector field A curve
C(?) with tangent equal to V.
where
Vector field set of 1st order ODEs. Solution
exists is unique in a region where vi(x) is
Lipschitzian (Choquet, p.95)
f X ? Y , where X, Y are Banach spaces, is
Lipschitzian in U ? X if ? k gt 0 s.t.
Uniqueness of solution ? integral curves
non-crossing ( except where V i 0 ? i )
Congruence the set of integral curves that
fills M.
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Integral curves
? V is not Lipschitzian in any neighborhood of
(0,0) since
as r ? 0
V is not Lipschitzian if any partial of V j does
not exist.
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2.13. Exponentiation of the Operator d/d?
Let M be C?. If the integral curve xi(?) of Y
d/d? is analytic, then
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2.14. Lie Brackets and Noncoordinate Bases
Natural vector field basis
with
Let

then
(Einstein's summation notation)
Lie bracket
involves Tp's of neighboring points
are noncoordinate basis vectors if
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Example of a noncoordinate basis (Exercise 2.1)
Coordinate grid xi is constant on the integral
curves xj of ?/?xj.
Integral curves of vector fields

? need not be constant on integral curves of d/dµ
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Ex 2.3 Jacobi Identity
Lie algebra (Additional) closure under Lie
bracket. Vector fields on U ? M is a linear space
closed under linear combinations of constant
coefficients. Lie algebra of vector fields on U
? M Closed under both. Invariances of M ?
Lie group (Chap 3)
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2.15. When Is A Basis A Coordinate Basis?
Let
be a basis for vector fields in an n-D region U
? M.
Then ?i is a coordinate (holonomic) basis for
U ?
Let xi be coordinates for U.
Proof for ?
Proof for ? (2-D case only)
Let
Task is to show that (a,ß) are good coordinates
and ( ?/?a, ?/?ß) are basis vectors.
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?
?
?
so that ( ?/?a, ?/?ß) are basis vectors if (a,ß)
are good coordinates
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defines a map (a,ß) ? ( x1, x2 )
If the map is invertible, then (a,ß) are good
coordinates. By the implicit function theorem,
this inverse exists ?
This is true since vectors A B are linearly
independent.
QED
General case
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1-Forms
Let X, Y be linear space over K. A mapping f
X ? Y is called linear if
The set L(X,Y) of all linear mappings from X to
Y is a linear space over K with
( vector addition )
( scalar multiplication )
The set L(X, K) is called the dual X of X.
Let Tx(M) be the dual to the tangent space Tx(M)
at x?M. Elements of Tx(M) are called cotangent
vectors, covariant vectors, covectors,
differential forms, or 1-forms. In contrast,
Elements of Tx(M) are called tangent vectors,
contravariant vectors, or simply vectors.
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Thus,
i.e., a 1-form is a linear function that takes a
vector to a number in K.
?
? one can define
( contraction , not to be confused with an inner
product )
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2.17. Examples of 1-Forms
Gradient of a function f is a 1-form denoted by
Matrix algebra Column matrix vector Row
matrix covector Matrix multiplication
contraction
Hilbert space in quantum mechanics Ket ???
vector Bra ? ? covector ? ? f ?
contraction / sesquilinear product
There is no natural way to associate vectors with
covectors. Doing so requires the introduction of
a metric or inner product.
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2.18. The Dirac Delta Function
Let C?1,1 be the set of all C? real functions
on interval ?1,1?R. ( C?1,1, ) is a
group. ( C?1,1, R ) is an ?-D vector
space. The dual 1-forms are called
distributions. E.g., the Dirac delta function
d(x-x0) is a distribution.
By definition, d(x-x0) maps a function f(x) to a
real number f(x0)
by
We shall avoid the abiguous notation
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By introducing the inner product
we can associate with vector g a unique 1-form G
by the condition
Since
one may say d(x?a) is a function in C?1,1 that
gives rise to the 1-form da .
However, calling d(x?a) is a function is not
mathematically correct. The delta function is
actually a Dirac measure (distribution of order
0). It can be shown that the Dirac measure cannot
be associated with a locally integrable function.
see Choquet, VI,B.
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2.19. The Gradient the Pictorial Representation
of a 1-Form
A 1-form field is a cross section of the
cotangent bundle T(M).
satisfying
The gradient of a function f M ? K is the 1-form
Let h(x) height at x. Then
? 1-form is a set of parallel planes
number of ? planes crossed by V
Steepest descend gradient vector can only be
defined in the presence of a metric.
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2.20. Basis 1-Forms Components of 1-Forms
Given a basis
for TP(M), there is a natural dual basis
for TP(M)
s.t.
Since
we have
?
Let
then
where
V is arbitrary ?
?
is indeed a basis for TP(M)
Coordinate basis
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2.21. Index Notation
Coordinate bases
? vector
Dual bases
? 1-form
Contraction
Einstein notation implied summation if a pair
of upper lower indices are denoted by the same
letter.
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2.22. Tensors Tensor Fields
A tensor F of type (NM) at P is a multi-linear
function that takes N 1-forms M vectors to a
number in K, i.e.,
Multi-linear means F is linear in each of its
arguments.
For a (21) tensor, we have
and
Linearity
etc
A vector is a (10) tensor, a 1-form is a (01)
tensor a function is a (00) tensor.
?
is a 1-form.
Operators or transformations on a linear space
are (11) tensors.
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2.2.3 Examples of Tensors
Matrix algebra Column vector is a (10) tensor,
row vector is a (01) tensor, matrix is a (11)
tensor.
Function space C-1,1 A differential operator
maps linearly a vector (function) into another
vector. ? it is a (11) tensor.
Elasticity Stress vector resultant force per
unit area across a surface Let the stress vector
over cube face S(k) with normal ek be F(k) , then
or
The normal to a plane is equivalent to a 1-form (
a set of parallel planes ) .
?t takes a 1-form into another vector ? it is a
(20) tensor called the stress tensor.
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2.24. Components of Tensors the Outer Product
Outer ( direct, or tensor ) product ? (NM)
tensor ? (PQ) tensor (NPMQ) tensor
Example
is a (20) tensor with
Caution
since
Reminder
? Cartesian product
The components of a (NM) tensor S are given by
so that
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2.25. Contraction
The summation of a pair of upper lower tensor
indices is called contraction.

1. Contraction of a (NM) with a (PQ) tensor gives
   a (NP?1MQ?1) tensor.
2. Contraction within a (NM) tensor gives a
   (N?1M?1) tensor.

Contraction is basis independent.
Proof for case A is rank (20) B rank (02)
Let C be the (11) tensor with components
?
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2.26. Basis Transformations
Tensor analysis A tensor is an object that
transforms like a tensor.
Consider
by
(? nonsingular )
with
Let
then
Hence
?
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( contravariant to ej )
?
( covariant to ej )
are basis independent
Coordinate transform
Coordinate basis
Dual basis
?
?
( not true for general bases )
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2.27. Tensor Operations On Components
Operations on components that result in another
tensor are called tensor operations. They are
necessarily basis independent.

• Examples
• Aij B ij C ij ? A B C
• a Aij ? a A
• Aij B kl C i kj l ? A?B C
• C i kk l D il ( contraction )

Equations containing only tensor operations on
tensors are called tensor equations. They are
necessarily basis independent.
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2.28. Functions Scalars
Scalar (00) tensor at a point on M. E.g., V
j ?j P. Scalar function on M (00) tensor field
on M. E.g., V j (x) ?j (x) V j (x) is a
function on M but not a scalar function. Note
It is possible to have a scalar function f(x)
s.t. f(x) V j (x) ? x. ? Cant tell whether
a function is scalar by looking at its values
alone.
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2.29. The Metric Tensor on a Vector Space
Given a real vector space X, one can define a
bilinear inner product
by
One can associate a (02) metric tensor g with ? by
g is symmetric since ? is
Matrix notation
Since ? is invertible, we can set ? O D, where
O is orthogonal D is diagonal.
?
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Since g is real-symmetric, it can be digonalized
with an orthogonal transform
? O can be chosen s.t.
Setting
?
g? is diagonal with elements ?1
Canonical form of g is diag( ?1, , ?1, 1, , 1
). The corresponding basis is orthonormal, i.e.,
Signature of g t Tr g (number of 1) ?
(number of ? 1)
Metric is positive-definite if t n ( all 1s
). Metric is negative-definite if t ?n ( all
?1s ). Definite metrics are called Euclidean.
Minkowski metric is indefinite.
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g I wrt the Cartesian basis for a Euclidean
space.
Transformations between different Cartesian bases
are orthogonal since
?
?
Canonical form of the Minkowski metric is
Corresponding bases are called Lorentz bases.
Transformations between different Lorentz bases
satisfy
They are called Lorentz transformations.
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A metric tensor associates a unique 1-form to
each vector via
where
?
i.e., g i j raises gi j lowers an index.
E.g.
In a metric space, tensors of ranks (NM) ,
(N1M?1), (N2M?2) , ., are all
equivalent. Theyre simply called tensors of
order NM.
In a Euclidean with a orthonormal basis, the
components of a vector its dual have the same
numerical value. ? In calculations, theres no
need to distiguish them.
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2.30. The Metric Tensor Field on a Manifold
A metric tensor field assigns continuously a
metric tensor g to TP(M) ? p? M. As already
stated, g must be (02), symmetric,
invertible. A metric is a high level structure
that allows the definition of distance,
curvature, metric connection, parallel transport,
. (see Chap 6) Reminder Lie derivatives
differential forms do not require a
metric. Continuity of g(p) ? canonical form of g
must be a constant ? p ? M. If one is free to
choose the basis for any TP(M), g can be made
canonical everywhere. However, these bases may
not all be coordinate ones (see Ex 2.14). One
exception is Rn with Euclidean metric Cartesian
coordinates (c.f. Ex 2.15-6).
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Arc-length ?l of a curve with tangent
is given by
If g is positive definite, then
If g is indefinite, then case ?l2 gt 0 is called
space-like ?l2 lt 0, time-like.
then gives the proper distance / proper time.
Case ?l2 0 is called light-like.
Note that
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2.31. Special Relativity
Minkowski spacetime R4 with metric of signature
2. Lorentz frames coordinates ( x0, x1, x2, x3
) ( ct, x, y, z ).
Metric tensor is in canonical form everywhere
Event intervals
are invariant under Lorentz transformations.
?s2 defines a pseudo-norm for the vector
Inner product is defined as