Submanifolds of products of space forms

Abstract

We give a complete classification of submanifolds with parallel second fundamental form of a product of two space forms. We also reduce the classification of umbilical submanifolds with dimension m ≥ 3 of a product Q n 1 k 1 × Q n 2 k 2 of two space forms whose curvatures satisfy k 1 + k 2 = 0 to the classification of m-dimensional umbilical submanifolds of codimension two of S n × R and H n × R. The case of S n × R was carried out in [13]. As a main tool we derive reduction of codimension theorems of independent interest for submanifolds of products of two space forms.MSC 2000: 53 B25, 53 C40.

Umbilical surfaces of products of space forms

“…Proof: If S vanishes everywhere on M 2 , then f is as in (i) by Lemma 8.1 in [3]. If ker S = {0} at some point x ∈ M 2 , then f is as in (ii), (iii) or (iv) by Lemma 4.…”

“…We will need the following result from [3] on reduction of codimension. In the statement, U and V stand for ker T and ker(I − T ), respectively.…”

“…We first consider the case (n 1 , n 2 ) = (3,2). This is the case in which T H = 0, and hence k 2 = 0 by (41).…”

“…In this case, the argument in one of the steps of the proof for the higher dimensional case (see Lemma 8.2 of [3]) does not apply, and requires more elaborate work. This is carried out in Lemma 4 below, which shows that the difficulty is due to the existence of new interesting families of examples in the surface case.…”

Pseudo-parallel surfaces of $$\mathbb {S}_c^n \times \mathbb {R}$$ S c n × R and $$\mathbb {H}_c^n \times \mathbb {R}$$ H c n × R

“…By δ i j we mean the Kronecker's Delta. From Mendonça and Tojeiro (2013) we have the following equations: Gauss:…”