The infinitesimally bendable Euclidean hypersurfaces

Abstract

The main purpose of this paper is to complete the work initiated by Sbrana in 1909 giving a complete local classification of the nonflat infinitesimally bendable hypersurfaces in Euclidean space.

Infinitesimal Bendings

“…Hence, from (17), (20) and (21) we have that T −T has the associated tensor β −β = Cα. A straightforward computation using the skew-symmetry of C and the Ricci equation yields that Cα and −(∇ ⊥ C) verify equations (6), (7) and (8). Then Proposition 3 gives E −Ẽ = −(∇ ⊥ C), and the proof now follows from Proposition 5.…”

“…The following result is Theorem 13 in [7] or Theorem 14.11 in [4]. Proof: In this case the tensor E vanishes.…”

“…On the contrary, the computation for the other two equations becomes really cumbersome outside the hypersurface case. For hypersurfaces this was done in [4], [7] and the result in coordinates for general codimension has been stated in [10].…”

“…(2). If the pair 0 = (β, E) satisfies (6), (7) and (8), then there is a unique infinitesimal bending T of f having (β, E) as associated pair.…”

“…The case of infinitesimally bendable Euclidean hypersurfaces has been initially considered around the beginning of the 20 th century, when the work of Sbrana [11] in 1908 stands out. A complete local parametric classification of the infinitesimally bendable hypersurfaces is due to Dajczer and Vlachos [7] who, in particular, showed that this class is much larger than the one of the isometrically bendable ones. The classification of complete infinitesimally bendable Euclidean hypersurfaces is due to Jimenez [9].…”