Differentiable Manifolds/What is a manifold?

Differentiable Manifolds
What is a manifold? Bases of tangent and cotangent spaces and the differentials → 

In this section, the important concepts of manifolds shall be introduced.

Charts, compatibility of charts, atlases and manifolds

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In this subsection, we define a manifold and all the things which are necessary to define it. It's a bit lengthy for a definition, but manifolds are such an important concept in mathematics that it's far more than worth it.

Definition 1.1:

Let   be a topological space, let   be a natural number, and let   be open. We call a function   a chart iff   is a homeomorphism and   is open in  .

Definition 1.2:

Let   be a topological space, let   be a natural number, let   be open, let   and   be two charts, and let  . We call the two charts compatible of class   iff either

 

or

both maps

 

and

 

are contained in  .

Definition 1.3:

Let   be a topological space, let  , let  , let  , where   is a set, be a set of open subsets of  , and let   be an according set of charts. We call the set   an atlas of class   of   iff both

  • for all  , there exists an   such that   and
  • for all   the charts   and   are compatible of class  .

Definition 1.4:

Let   be a topological space, let   and let  . Together with an atlas   of   of class  , we call   a  -dimensional manifold of class  .

Differentiable functions on manifolds

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In this subsection, we shall define what differentiable maps, which map from a manifold or to a manifold or both, are.

Definition 1.5:

Let   be a  -dimensional manifold of class  , let   be an atlas of it, and let  , and let   be open. We call a function   differentiable of class   iff for all   the function

 

is contained in  .

We write   for the set of all real-valued differentiable functions of class   from any open subset of   to  . Further, we write   for the set of continuous functions from any open subset of   to   (remember that both   and   are topological spaces, which is why continuity is defined for functions from one of them to the other).

On this set  , we define addition and multiplication as follows: Let   be open and  ,   be two differentiable functions of class  . We define

 
 
 

and, if   is never zero,

 

Instead of writing  , we will in the following write  ; just omitting the dot. This is often also done for the multiplication of variables (for instance   stands for   if  ).

Definition 1.6:

Let   be a  -dimensional manifold of class  , let   be an atlas of it, let   and let  . We call a function   a differentiable curve of class   iff for all   the function

 

is contained in  .

Definition 1.7:

Let   be manifolds of dimensions   respectively, let   and let  ,   be atlases of   and   respectively. We call a function   differentiable of class   iff for all   and all   either

 

or the function

 

is contained in  .

Tangent vectors, tangent spaces and tangent bundles

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Tangents, in the classical sense, are lines which touch a geometrical object at exactly one point. The following definition of a tangent of a manifold, in this context called tangent vector to a manifold, is somewhat strange.

Definition 1.8:

Let   be a manifold of class  , and let  . A tangent vector at   is a function   such that for all   and   the following three rules hold:

  1.   whenever   and   are both defined at   (linearity)
  2.   whenever   and   are both defined at   (product rule)
  3. if   is not defined at   (i. e.   is a function from   to   such that  ), then  .

Definition 1.9:

Let   be a manifold, and let  . The tangent space of   in  , which we shall denote by  , is defined to be the vector space of all tangent vectors at   with the scalar-vector multiplication

 ,

the vector-vector addition

 

and the zero element

 .

Definition 1.10:

Let   be a manifold. The tangent bundle of  , denoted by  , is defined as follows:

 

Cotangent vectors, cotangent spaces and cotangent bundles

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Definition 1.11:

Let   be a manifold of class  , and let  . A linear function from   to   is called cotangent vector at  . One standard symbol for a cotangent vector at   is  .

Definition 1.12:

Let   be a manifold, and let  . The cotangent space of   in  , which we shall denote by  , is defined to be the vector space of all cotangent vectors at   with the scalar-vector multiplication

 ,

the vector-vector addition

 

and the zero element

 .

Definition 1.13:

Let   be a manifold. The cotangent bundle of  , denoted by  , is defined as follows:

 

Tensors and the tensor product

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Definition 1.14:

Let   be a vector space,   its dual space and let  . We call a multilinear function

 

a   tensor.

Definition 1.15: Let   be a vector space, let   be its dual space, let  , let   be a   tensor and let   be a   tensor. The tensor product of   and  , denoted by  , is defined to be the   tensor given by

 

We denote the set of all tensors with respect to   by  .

Sources

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  • Torres del Castillo, Gerardo (2012). Differentiable Manifolds. Boston: Birkhäuser. ISBN 978-0-8176-8271-2.
  • Lang, Serge (2002). Introduction to Differentiable Manifolds. New York: Springer. ISBN 0-387-95477-5.
  • Rudolph, Gerd; Schmidt, Matthias (2013). Differential Geometry and Mathematical Physics. Netherlands: Springer. ISBN 978-94-007-5345-7.
Differentiable Manifolds
What is a manifold? Bases of tangent and cotangent spaces and the differentials →