Surface geometry
This notebook illustrates how to use the Atlas package to solve problems in elementary differential geometry. As an example we find the
geometry of the following surfaces:
torus, helicoid and
abstract revolution surface.
What we do?
- We assume that the geometry is induced by the corresponding embedding of a surface into the flat Euclidian 3-dimensional space.
- Thus we construct the corresponding embedding and calculate the following quantities: first fundamental form (metric tensor field), second fundamental form and field of mean curvature vectors.
- Besides that we calculate the connection, curvature, curvature tensor field (Riemann tensor field), Ricci tensor field and Ricci scalar function which is proportional to scalar curvature.
| Connection[id] | id-variable-connection identifier. |
| 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:
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[8]= |  |
Declare 1-forms for the space coframe:
| Out[9]= |  |
Declare vectors for the space frame:
| Out[10]= |  |
Declare coframe on the space:
| Out[11]= |  |
Declare frame on the space:
| Out[12]= |  |
Declare flat metric on the space:
| Out[13]= |  |
Calculate connection of the metric:
| Out[14]= |  |
Torus
Define the torus as a manifold:
| Out[15]= |  |
Declare constants a, b, c:
| Out[16]= |  |
Declare 1-forms for torus coframe:
| Out[17]= |  |
Declare vectors for torus frame:
| Out[18]= |  |
Declare coframe on the torus:
| Out[19]= |  |
Declare frame of the surface:
| Out[20]= |  |
Declare mapping of the torus into
R2:
| Out[21]= |  |
| Out[22]= |  |
Calculate metric on the torus using
Pullback- operator:
| Out[23]= |  |
Calculate invariants of the mapping:
| Out[24]= |  |
Extract second fundamental form as a table:
| Out[25]= |  |
Construct second fundamental form as corresponding tensor field (
CircleTimes - is tensor product operator):
| Out[26]= |  |
Extract mean curvature vector:
| Out[27]= |  |
Calculate mean curvature:
| Out[28]= |  |
Use differential operator Pushforward to express tangent vectors on the torus
Wi by total space frame vectors
j:
| Out[29]= |  |
| Out[30]= |  |
Verify that mean curvature vector field is normal to the tangent space of the torus:
| Out[31]= |  |
| Out[32]= |  |
Calculate connection of the embedding:
| Out[33]= |  |
Calculate curvature of the embedding:
| Out[34]= |  |
Calculate Riemann tensor field:
| Out[35]= |  |
Calculate Ricci tensor field:
| Out[36]= |  |
Calculate Ricci scalar curvature:
| Out[37]= |  |
Helicoid
Define the surface as a manifold:
| Out[38]= |  |
Declare 1-form for surface coframe:
| Out[39]= |  |
Declare vectors for surface frame:
| Out[40]= |  |
Declare coframe on the surface:
| Out[41]= |  |
Declare frame of the surface:
| Out[42]= |  |
Declare mapping of the surface into
R3:
| Out[43]= |  |
| Out[44]= |  |
One can also calculate metric induced on the surface by the mapping:
| Out[45]= |  |
Calculate invariants of the mapping:
| Out[46]= |  |
It is well known that helicoid is a minimal surface. Thus nothing surprising about zero mean curvature vector field.
Calculate connection of the embedding:
| Out[47]= |  |
Calculate curvature of the embedding:
| Out[48]= |  |
Calculate Riemann tensor field:
| Out[49]= |  |
Calculate Ricci tensor field:
| Out[50]= |  |
Calculate Ricci scalar curvature:
| Out[51]= |  |
Abstract revolution surface
Define the surface as a manifold:
| Out[52]= |  |
| Out[53]= |  |
Declare 1-form for surface coframe:
| Out[54]= |  |
Declare vectors for surface frame:
| Out[55]= |  |
Declare coframe on the surface:
| Out[56]= |  |
Declare frame of the surface:
| Out[57]= |  |
Declare mapping of the surface into
R3:
| Out[58]= |  |
| Out[59]= |  |
One can also calculate metric induced on the surface by the mapping:
| Out[60]= |  |
Calculate invariants of the mapping:
| Out[61]= |  |
Calculate connection of the embedding:
| Out[62]= |  |
Calculate curvature of the embedding:
| Out[63]= |  |
Calculate Riemann tensor field:
| Out[64]= |  |
Calculate Ricci tensor field:
| Out[65]= |  |
Calculate Ricci scalar curvature:
| Out[66]= |  |
- Lie derivative
- Exterior product
- Tensor product
- Hodge operator
- Covariant derivative
