Definition by Tangent Curves
Tangent Curve
Fix an
-manifold and a point . A tangent curve based at is a smooth map sending , where is an open neighbourhood of .
First Order Agreement
On an
-manifold , two tangent curves and based at are said to agree to first order if there exists a chart around such that We write for the map .
Lemma
If
and satisfy , that is they agree to first order, then and agree to first order in any chart around .
Proof Suppose
Corollary
Agreement to first order is an equivalence relation.
Proof Clear enough as '
Smooth Path Space
The smooth path space to some manifold
at point , denoted is the set of all tangent curves based at modulo the equivalence relation of first order agreement: And we introduce the vector operations on by selecting a chart around , and define . Clearly is a bijection, and now we can define the vector space operations on by
Definition by Vectors in
Tangent Space
For
we define to be the set of pairs where is a chart in the atlas for with , and is an element of , modulo the equivalence relation which identifies a pair with a pair if and only if maps to under the derivative of the transition map between the two charts: We denote such an equivalence class by .
Remark
We think of a vector as being an ‘arrow’ telling us which way to move inside the manifold. This information on which way to move is encoded by viewing the motion through a chart
, and seeing which way we move ‘downstairs’ in the chart (this corresponds to a vector in according to the usual notion of a velocity vector). The equivalence relation just removes the ambiguity of a choice of chart through which to follow the motion.
Definition
Definition by Smooth Functions
Derivation Space
Denote
as the set of real smooth functions whose domain includes some open neighborhood of . A derivation at a point is a map satisfying for all and the two conditions: And we define to be the set of all derivations at .
Derivations are directional derivatives
The archetypal example of a derivation is the directional derivative of a function along a curve: Given a smooth path
with , we can define Indeed, all derivations are of this form.
Three Definitions of Tangent Space are Equivalent
There are isomorphisms between the three spaces
, and .
Proof
Remark
These are three characterizations of the tangent space to a manifold at a point. From now on, we will use the notation
to denote the tangent space at , and treat each vector in as whatever definition is most convenient.
Proposition
Suppose
is a smooth -manifold. The tangent space has dimension for all .
Proof It suffices to show that the differential of a local chart
Standard Basis of Tangent Space
Now suppose the chart
around is given. Then one can define an ordered basis for as derivations: where each partial derivative operator (at ) is defined as
Proof Since we know the tangent space has dimension
The Differential of a Smooth Function
Differential
Let
be a smooth function. Then we define the differential of at point to the linear function on given by In general, the differential of a smooth map between two manifolds at , written is defined as
- For the tangent curve definition,
. - For the chart definition,
. - For the derivation definition,
.
We shall check that the above is well-defined, and all three definitions are equivalent:
Tangent Bundle
Tangent Bundle
A tangent bundle is the disjoint union of the tangent spaces of a manifold
:
e.g. Consider
Smooth Structure on Tangent Bundle
Given a smooth
-manifold , with an atlas . Then has a canonical structure of dimension with its atlas given by
Proof The above construction gives a unique topology on
e.g.
Proposition
is a vector bundle over .
Proof The bundle projection is the map