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Classification of Rank-One Submanifolds
1Institute of Discrete Mathematics and Geometry, TU Wien, Wiedner Hauptstraße 8-10/104, 1040 Vienna, Austria.
Results in Mathematics
|August 22, 2023
Summary
We introduce a "degree" function for ruled submanifolds to quantify non-cylindrical behavior. Constant degree implies singularities are confined to a specific "striction" submanifold, aiding classification.
Area of Science:
- Differential Geometry
- Submanifold Theory
Background:
- Ruled submanifolds are fundamental geometric objects.
- Understanding their singularities is crucial for classification.
Purpose of the Study:
- To introduce a new invariant, the 'degree', for ruled submanifolds.
- To classify flat and ruled submanifolds without planar points (rank-one submanifolds).
Main Methods:
- Associating an integer-valued 'degree' function to each ruled submanifold.
- Analyzing the geometric implications of a constant degree.
Main Results:
- A constant degree 'd' restricts singularities to a d-dimensional 'striction' submanifold.
- This extends the classification of developable surfaces to rank-one submanifolds.
- Rank-one submanifolds are shown to be composed of cylindrical, conical, and tangent regions.
Conclusions:
- The 'degree' function provides a powerful tool for analyzing ruled submanifolds.
- A comprehensive classification of rank-one submanifolds is established.
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