Path: blob/main/15Manifold_Pullback.ipynb
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15. Pulback of tensor fields
This notebook is part of the Introduction to manifolds in SageMath by Andrzej Chrzeszczyk (Jan Kochanowski University of Kielce, Poland).
Recall from notebook 7, that every smooth map between two smooth manifolds defines a pullback map by
Example 15.1
If we denote by the coordinate change defined by on the open set then for a scalar function of variables the pullback is the function , defined on .
In the special case , the pullback is the function on :
Pullback of covariant tensor fields
If is a smooth map between smooth manifolds, one can extend the pullback operation to a map by
ParseError: KaTeX parse error: Undefined control sequence: \label at position 105: …dψ_p u_{kp} ), \̲l̲a̲b̲e̲l̲{}\tag{15.1} \e…for From the linearity of it follows the multilinearity of .
If , then the differential of is a tensor field of type , consequently from (15.1) it follows
for
From the definitions of and we have
We have checked that for a smooth map and
ParseError: KaTeX parse error: Undefined control sequence: \label at position 39: …f = d(ψ^∗ f ). \̲l̲a̲b̲e̲l̲{}\tag{15.2} \e…Example 15.2
Compute the pullback of differential of the scalar function from Example 15.1.
First we compute the left hand side of (15.2):
Here is the right hand side of (15.2):
Example 15.3
Compute pullback of a scalar function and its differential for general mapping between two-dimensional manifolds and
Example 15.4
Compute the pullback of 1-form under the map from the previous example.
Shorter result can be obtained omitting functions arguments.
Example 15.5
Compute the pullback of a tensor field of (0,2)-type on a two dimensional manifold under from Example 15.3.
Basic properties of pullback
For a smooth map tensor fields and on of type and constants we have
for . Therefore we have the following linearity result.
Let be a smooth map. If and are tensor fields of type on and , then
ParseError: KaTeX parse error: Undefined control sequence: \label at position 51: …^∗ t + bψ^∗ s. \̲l̲a̲b̲e̲l̲{}\tag{15.3} \e…For as before, and , we have
Thus ParseError: KaTeX parse error: Undefined control sequence: \label at position 48: …^∗ f )(ψ^∗ t).
\̲l̲a̲b̲e̲l̲{}\tag{15.4}
\e…
If is smooth, and are tensor fields on of types and respectively, we have
so
ParseError: KaTeX parse error: Undefined control sequence: \label at position 51: … t) ⊗ (ψ^∗ s). \̲l̲a̲b̲e̲l̲{}\tag{15.5} \e…If are smooth manifolds, and are smooth maps, then
To prove this relation let us note that
Pullback of covariant tensor fields in components
If is a tensor field of type on , given locally by , then the pullback of under is given by
Since , where are coordinates on , we have
The same formula can be written:
So to obtain , just replace in
and
Example 15.6
Consider the covariant tensor field in and its pullback under the embedding defined by
metric
is a predefined covariant tensor field of type (0,2) in
The pullback of g:
Example 15.7
We can repeat the previous example in higher dimensions. For example we can compute the pullback of under the embedding
metric
is a predefined covariant tensor field of type (0,2) in
Pullback of g:
Since the differential k-forms are special cases of covariant tensor fields, the pullback operation can be applied to such forms.
Example 15.8
Consider the same embedding as in Example 15.6, defined by and compute the pullbacks
More examples of pullbacks of k-forms can be found in the next notebook.
What's next?
Take a look at the notebook Exterior derivative.