**Notation**

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**Chapter 1**

*1. Differentiable Manifolds*

- Let be a topological space. A
on is a set of tansformations satisfying the following axioms:__pseudogroup of transformations__- Each is a homeomorphism of an open subset of onto the open subset of .
- If , then the restriction of to an arbitrary open subset of the domain of is in .
- Let where each is an open set of . A homeomorphism of onto an open set of belongs to if the restriction of to is in for every .
- For every open set of , the identity transformation of is in .
- If , then .
- If is a homeomorphism of onto and is a homeomorphism of onto and if is non-empty, then the homeomorphism of onto is in .

- A mapping of an open set of into is said to be of
, , if is continuously times differentiable.__class__- The
, denoted , is the set of homeomorphisms of an open set of onto an open set of such that both and are of class .__pseudogroup of transformations of class__ - The colleciton of those whose Jacobians are positive everywhere is a subpseudogroup of called the
and is denoted .__pseudogroup of orientation-preserving transformations of class of__

- The
- An
of a topological space compatible with a pseudogroup is a family of pairs called__atlas__which satisfy the following:__charts__- Each is an open set of and
- Each is a homeomorphism of onto an open set of .
- Whenever is nonempty, the mapping of onto is an element of .

- A
atlas of compatible with is an atlas of compatible with which is not contained in any other atlas of compatible with .__complete__ - A
of class is a Hausdorff space with a fixed complete atlas compatible with . An__differentiable manifold__of class is a Hausdorff space with a fixed complete atlas compatible with .__oriented differentiable manifold__ - An
is a differentiable structure of class obtained from an oriented differentiable structure of class .__orientable differentiable structure__ - An
of a structure is a chart which belongs to the fixed complete atlas defining the structure.__allowable chart__ - Given an allowable chart of an -dimensional manifold of class , the system of functions defined on is called a
in and is called a__local coordinate system__.__coordinate neighborhood__ - Given two manifolds and of class , a mapping is said to be
, , if, for every chart of and every chart of such that , the mapping of into is differentiable of class .__differentiable of class__- A maping of class is referred to as a
.__(differentiable) mapping__

- A maping of class is referred to as a
- A
in is a differentiable mapping of class of a closed interval of into , that is, a restriction of a differentiable mapping of class of an open interval containing into .__differentiable curve of class__ - Given a manifold and a point , let be the algebra of differentiable functions of class defined in a neighborhood of .
- Let be a curve of class , , such that . The
is a mapping defined by , i.e. is the derivative of in the direction of the curve at .__vector tangent to the curve at__ - The set of tangent vectors at is called the
of at and is denoted by or .__tangent space__- Given a tangent vector , the -tuple is called the
of the vector with respect to the local coordinate system .__components__

- Given a tangent vector , the -tuple is called the
- A
on a manifold is an assignment of a vector to each point on . In particular, if is a differentiable function on , then is a function on defined by .__vector field__- A vector field is called
if is differentiable for every differentiable function .__differentiable__ - Givne a local coordinate system , a vector field may be expressed as , where are again called
of with respect to .__components__

- A vector field is called
- Let be the set of all differentiable vector fields on .
- Given , define the
of as a mapping from the ring of functions on to itself by .__bracket__ - For , the dual vector space of is called the space of
at .__covectors__- An assignment of a covector at each point is called a
or a__1-form__.__differential form of degree 1__ - The functions (defined in the neighborhood of ) are called the
of the 1-form (defined in a neighborhood of ) with respect to .__components__ - The 1-form is called
if each is differentiable.__differentiable__

- An assignment of a covector at each point is called a
- For each function on , the
of at is defined by for where denotes the value of the first entry on the second entry as a linear functional on .__total differential__ - Recall that an exterior algebra over a vector space is, roughly, an algebra whose product is the exterior product . Let be the exterior algebra over . An
is an assignment of an element of degree in to each point of .__-form__- In terms of a local coordinate system , can be expressed uniquely as .
- The -form is called
if the components are all differentiable.__differentiable__

- Denote by the collection of all differentiable -forms on for each . Let .
on can be characterized as follows:__Exterior Differentiation__- is an -linear mapping of into itself such that .
- For a function , is the total differential.
- If and , then .

In terms of a local coordinate system, if , then

- Given a mapping of a manifold into another manifold , the
at of is the linear mapping of into defined as follows: For each , choose a curve in such that is the vector tangent to at . Then is the vector tangent to the curve at . When it is necessary to specify the point , the notation is used.__differential__- The transpose of is a linear mapping of into .

- For any -form on , define the -form on by for .
- Note that exterior differentiation commutes with so that .

- A mapping of into is said to
at if the dimension of is .__be of rank__- If the rank of at is equal to , is injective and .
- If the rank of at is equal to , is surjective and .

- A mapping of into is called an
if is injective for every point in . Here, we say that is__immersion__in by or that is an__immersed__of .__immersed submanifold__- When an immersion is inejctive, it is called an
of into and it is said that is a(n)__imbedding__of .__(imbedded) submanifold__

- When an immersion is inejctive, it is called an
- An open subset of a manifold considered as a submanifold of in a natural manner is called an
.__open submanifold of__ - A
of a manifold onto another manifold is a homeomorphism such that both and are differentiable.__diffeomorphism__- A diffeomorphism of onto itself is called a
of .__(differentiable) transformation__

- A diffeomorphism of onto itself is called a
- A transformation of induces an automorphism of the algebra of differential forms on and, in particular, an automorphism of the algebra of functions on : for , .

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