Ellipse of normal curvature

A geometrical construction characterizing the distribution of the curvatures at a certain point on a regular surface $ M ^ {2} $ in the $ n $- dimensional Euclidean space $ E ^ {n} $. Let $ P $ be a point on a surface $ M ^ {2} $ and let $ N _ {\mathbf l } $ be the $ ( n - 1) $- dimensional subspace containing the normal space $ N $ to $ M ^ {2} $ at $ P $ and the tangent to $ M ^ {2} $ at $ P $ in the direction $ \mathbf l $. The section $ \gamma _ {\mathbf l } $ of $ M ^ {2} $ by $ N _ {\mathbf l } $ is called a normal section at $ P $. The vector $ d ^ {2} \gamma _ {\mathbf l } /ds ^ {2} $, lying in $ N $, where $ s $ is the natural parameter on $ \gamma _ {\mathbf l } $, is called the vector of normal curvature of $ M ^ {2} $ in the direction $ \mathbf l $. The end points of the vectors of normal curvature form the ellipse of normal curvature.

For a two-dimensional surface $ M ^ {2} $ with non-zero Gaussian curvature in $ E ^ {n} $ to lie in a certain three-dimensional subspace $ E ^ {3} $ it is necessary and sufficient that its ellipse of normal curvature at all points $ P $ degenerates to a segment passing through $ P $( see ).

Similarly one defines the indicatrix of curvature for a submanifold $ M ^ {m} $ of arbitrary dimension $ m $. It is an $ ( m - 1) $- dimensional algebraic surface of order $ 2 ^ {m-} 1 $. The vectors of normal curvature form a cone which, together with the tangent space to $ M ^ {m} $, determines a subspace $ E ^ {m _ {1} } $, the so-called domain of curvature of $ M ^ {m} $ at $ P $. The dimension $ m _ {1} $ of this subspace satisfies

$$ m _ {1} \leq \ { \frac{m ( m + 3) }{2} } ,\ \ m _ {1} \leq n. $$

Points at which $ m _ {1} = m + 1 $ are called axial, those at which $ m _ {1} = m + 2 $— planar, and those at which $ m _ {1} = m + 3 $— spatial. Sometimes one considers for submanifolds of large codimension the Dupin indicatrix, the construction of which is completely analogous to that of the Dupin indicatrix for a surface in three-dimensional space.

References

[1]	J.A. Schouten, D.J. Struik, "Einführung in die neueren Methoden der Differentialgeometrie" , 2 , Noordhoff (1935)
[2a]	Yu.A. Aminov, "Torsion of two-dimensional surfaces in Euclidean spaces" Ukrain. Geom. Sb. , 17 (1975) pp. 3–14 (In Russian)
[2b]	Yu.A. Aminov, "An analogue of Ricci's condition for a minimal variety in a Riemannian space" Ukrain. Geom. Sb. , 17 (1975) pp. 15–22; 144 (In Russian)

References

[a1]	M. do Carmo, "Differential geometry of curves and surfaces" , Prentice-Hall (1976)
[a2]	B. O'Neill, "Elementary differential geometry" , Acad. Press (1966)
[a3]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1–2 , Interscience (1969) pp. Chapt. 7
[a4]	M. Spivak, "A comprehensive introduction to differential geometry" , 3 , Publish or Perish pp. 1–5