Winding line on a torus
This notebook illustrates how to use the Atlas package to solve problems in elementary differential geometry. As an example we calculate the
curvatures and
moving frame of a winding line on a torus. This line is embedded into the Euclidian space, on the other hand the line is embedded into the torus. We assume that the torus is embedded into the Euclidian space and the metric tensor field of the torus is just the induced metric of the embedding.
What we do?
- In the first case we calculate the moving frame: principal normal, binormal and tangent vector fields for the embedded curve.
- We also calculate first and second curvatures of the embedded curve which are known as curvature and torsion of the curve.
- In the second case we calculate the moving frame: principal normal and tangent vector fields for the embedded curve.
- Besides that we calculate first curvature of the embedded curve.
- As an intermediate result we obtain the first fundamental forms induced by the corresponding embedding on the curves and the torus (metric tensor field for the torus and squared differential of the curve's arc for curves).
- Additionally we obtain the second fundamental form and mean curvature vector field of the embedded torus.
| Forms[f1→n,f2→k,...,fi→p] | fi→p-rules where fi-form identifier and p is a variable or an integer-the form's degree. |
| Vectors[v1,v2,...,vi,...,vn] | vi - vector identivier. |
| Coframe[id1→ expr1,id2→ expr2,...idn→ exprn] | id - identifier for indexed variable - the coframe 1-forms,
n - dimension of working manifold (a variable or integer),
idi→ expri - rule where idi is indexed variable - coframe 1-form and expri is decomposition of the 1-form on exact 1-forms. |
| Invariants[f] | f - mapping identifier. |
Necessary functions.
Solution:
Just for right simplification:
Description of the total space R3
First of all we have to describe the space we are working in. The space is 3-dimensional Euclidean (flat) space. To define the space we declare domain, forms, vectors, coframe, frame, flat metric and calculate connection (it is equal to zero of course).
Define Euclidean space as a manifold:
| Out[9]= |  |
Declare 1-forms for the space coframe:
| Out[10]= |  |
Declare vectors for the space frame:
| Out[11]= |  |
Declare coframe on the space:
| Out[12]= |  |
Declare frame on the space:
| Out[13]= |  |
Declare flat metric on the space:
| Out[14]= |  |
Calculate connection of the metric:
| Out[15]= |  |
Description of the curve C (winding line on a torus)
Define the curve as a manifold:
| Out[16]= |  |
Declare constants a, b and m, n:
| Out[17]= |  |
Declare 1-form for curve's coframe:
| Out[18]= |  |
Declare vectors for curve's frame:
| Out[19]= |  |
Declare coframe on the curve:
| Out[20]= |  |
Declare frame of the curve:
| Out[21]= |  |
Mapping 
:C
R3 and its invariants (metric, moving frame, curvature, torsion)
Declare mapping of the curve into
R3:
| Out[22]= |  |
| Out[23]= |  |
One can also calculate metric induced on the curve by the mapping:
| Out[24]= |  |
It is obvious that the metric gives
squared differential of the curve's arc:
d(l)2=(m2a2+2m2a b cos(n
)+m2b2cos(n)2+b2n2)d()2Calculate invariants of the mapping:
Extract tangent normalized vector field:
| Out[26]= |  |
Extract principal normal normalized vector field:
Extract binormal normalized vector field:
Extract curvature of the curve:
| Out[29]= |  |
Extract torsion of the curve:
| Out[30]= |  |
Let us check the "orthonormality" of the moving frame vectors. To do this we use metric tensor field
gR:
| Out[31]= |  |
| Out[32]= |  |
| Out[33]= |  |
| Out[34]= |  |
| Out[35]= |  |
| Out[36]= |  |
Description of the torus T2
Define the torus as a manifold:
| Out[37]= |  |
Declare 1-form for the torus coframe:
| Out[38]= |  |
Declare vectors for the torus frame:
| Out[39]= |  |
Declare coframe on the torus:
| Out[40]= |  |
Declare frame of the torus:
| Out[41]= |  |
Embedding f:TR3 and induced metric calculation
Declare mapping of the torus into
R3:
| Out[42]= |  |
| Out[43]= |  |
Calculate metric (induced by the embedding) on the torus using
Pullback operator:
| Out[44]= |  |
Calculate connection on the torus:
| Out[45]= |  |
Out[46]//MatrixForm= |
| |  |
We can calculate the corresponding invariants of the mapping. The invarians are:
second fundamental form and field of
mean curvature vectors. We do not need these quantities anyway, but it's hard to keep from the calculation:
| Out[47]= |  |
Embedding 
:CT2 and its invariants (metric, moving frame, curvature)
Declare mapping of the curve into
T2:
| Out[48]= |  |
Calculate metric (induced by the embedding) on the curve using
Pullback operator:
| Out[49]= |  |
Calculate invariants of the mapping:
| Out[50]= |  |
Extract tangent normalized vector field:
| Out[51]= |  |
Extract principal normal normalized vector field:
| Out[52]= |  |
Extract curvature of the curve:
| Out[53]= |  |
Let us check the "orthonormality" of the moving frame vectors. To do this we use metric tensor field
gT, but in this case the metric is not flat and its components (in any coframe) are verify from one torus point to another:
| Out[54]= |  |
Thus we have to make calculations on the curve C:
| Out[55]= |  |
| Out[56]= |  |
| Out[57]= |  |
| Out[58]= |  |
- Lie derivative
- Exterior product
- Tensor product
- Hodge operator
- Covariant derivative
