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Complete submanifolds of Euclidean space with codimension two

Fernando Manfio University of São Paulo

Joint work with Cleidinaldo Silva – UFPI

Symmetry and Shape Celebrating the 60th birthday of Prof. J. Berndt

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 1 / 10

Motivation

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Motivation

Classical problem in submanifold theory: study of isometric immersions f : Mn → Rn+k of a complete Riemannian manifold under the action of a closed Lie subgroup G ⊂ Iso(M).

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 2 / 10

Motivation

Classical problem in submanifold theory: study of isometric immersions f : Mn → Rn+k of a complete Riemannian manifold under the action of a closed Lie subgroup G ⊂ Iso(M). Goal: To classify isometric immersions f : Mn → Rn+2 of a compact Riemannian manifold Mn of cohomogeneity one under the action of a closed Lie subgroup G ⊂ Iso(M) such that the principal

• rbits are umbilic hypersurfaces in Mn.

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 2 / 10

The hypersurface case

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 3 / 10

The hypersurface case

Theorem (Kobayashi, Trans. Am. Math. Soc., 1958):

Let f : Mn → Rn+1 be an isometric immersion of a compact homogeneous Riemannian manifold, i.e., Iso(M) acts transitively on M. Then f embeds Mn as a round sphere.

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 3 / 10

The hypersurface case

Theorem (Kobayashi, Trans. Am. Math. Soc., 1958):

Let f : Mn → Rn+1 be an isometric immersion of a compact homogeneous Riemannian manifold, i.e., Iso(M) acts transitively on M. Then f embeds Mn as a round sphere. Extension of Kobayashi’s theorem to the noncompact case:

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 3 / 10

The hypersurface case

Theorem (Kobayashi, Trans. Am. Math. Soc., 1958):

Let f : Mn → Rn+1 be an isometric immersion of a compact homogeneous Riemannian manifold, i.e., Iso(M) acts transitively on M. Then f embeds Mn as a round sphere. Extension of Kobayashi’s theorem to the noncompact case:

Theorem (Nagano-Takahashi, J. Math. Soc. Japan, 1960):

Let f : Mn → Rn+1 be an isometric immersion of a connected homogeneous Riemannian

• manifold. Then f(M) is isometric to the product Sk × Rn−k.

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 3 / 10

The hypersurface case

Theorem (Kobayashi, Trans. Am. Math. Soc., 1958):

Let f : Mn → Rn+1 be an isometric immersion of a compact homogeneous Riemannian manifold, i.e., Iso(M) acts transitively on M. Then f embeds Mn as a round sphere. Extension of Kobayashi’s theorem to the noncompact case:

Theorem (Nagano-Takahashi, J. Math. Soc. Japan, 1960):

Let f : Mn → Rn+1 be an isometric immersion of a connected homogeneous Riemannian

• manifold. Then f(M) is isometric to the product Sk × Rn−k.

Theorem (Ros, J. Differ. Geom., 1988):

Let f : Mn → Rn+1 be an isometric immersion of a compact Riemannian manifold. If the scalar curvature of Mn is constant, then f(M) is isometric to a sphere.

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 3 / 10

The hypersurface case

Theorem (Kobayashi, Trans. Am. Math. Soc., 1958):

Let f : Mn → Rn+1 be an isometric immersion of a compact homogeneous Riemannian manifold, i.e., Iso(M) acts transitively on M. Then f embeds Mn as a round sphere. Extension of Kobayashi’s theorem to the noncompact case:

Theorem (Nagano-Takahashi, J. Math. Soc. Japan, 1960):

Let f : Mn → Rn+1 be an isometric immersion of a connected homogeneous Riemannian

• manifold. Then f(M) is isometric to the product Sk × Rn−k.

Theorem (Ros, J. Differ. Geom., 1988):

Let f : Mn → Rn+1 be an isometric immersion of a compact Riemannian manifold. If the scalar curvature of Mn is constant, then f(M) is isometric to a sphere. Cohomogeneity one:

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 3 / 10

The hypersurface case

Theorem (Kobayashi, Trans. Am. Math. Soc., 1958):

Let f : Mn → Rn+1 be an isometric immersion of a compact homogeneous Riemannian manifold, i.e., Iso(M) acts transitively on M. Then f embeds Mn as a round sphere. Extension of Kobayashi’s theorem to the noncompact case:

Theorem (Nagano-Takahashi, J. Math. Soc. Japan, 1960):

Let f : Mn → Rn+1 be an isometric immersion of a connected homogeneous Riemannian

• manifold. Then f(M) is isometric to the product Sk × Rn−k.

Theorem (Ros, J. Differ. Geom., 1988):

Let f : Mn → Rn+1 be an isometric immersion of a compact Riemannian manifold. If the scalar curvature of Mn is constant, then f(M) is isometric to a sphere. Cohomogeneity one:

Theorem (Podestà-Spiro, Ann. Global Anal. Geom., 1995):

Let Mn be a compact Riemannian manifold under the action of a closed Lie subgroup G ⊂ Iso(M) with cohomogeneity one, and let f : Mn → Rn+1 be an isometric immersion.

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 3 / 10

The hypersurface case

Theorem (Kobayashi, Trans. Am. Math. Soc., 1958):

Let f : Mn → Rn+1 be an isometric immersion of a compact homogeneous Riemannian manifold, i.e., Iso(M) acts transitively on M. Then f embeds Mn as a round sphere. Extension of Kobayashi’s theorem to the noncompact case:

Theorem (Nagano-Takahashi, J. Math. Soc. Japan, 1960):

Let f : Mn → Rn+1 be an isometric immersion of a connected homogeneous Riemannian

• manifold. Then f(M) is isometric to the product Sk × Rn−k.

Theorem (Ros, J. Differ. Geom., 1988):

Let f : Mn → Rn+1 be an isometric immersion of a compact Riemannian manifold. If the scalar curvature of Mn is constant, then f(M) is isometric to a sphere. Cohomogeneity one:

Theorem (Podestà-Spiro, Ann. Global Anal. Geom., 1995):

Let Mn be a compact Riemannian manifold under the action of a closed Lie subgroup G ⊂ Iso(M) with cohomogeneity one, and let f : Mn → Rn+1 be an isometric immersion. Then f(M) is a rotational hypersurface if and only if the principal orbits are umbilics.

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 3 / 10

The hypersurface case: more general examples

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The hypersurface case: more general examples

cohomogeneity two compact subgroup G ⊂ SO(n + 1),

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 4 / 10

The hypersurface case: more general examples

cohomogeneity two compact subgroup G ⊂ SO(n + 1), γ curve that is either contained in the interior of Rn+1/G or meets its boundary orthogonally,

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 4 / 10

The hypersurface case: more general examples

cohomogeneity two compact subgroup G ⊂ SO(n + 1), γ curve that is either contained in the interior of Rn+1/G or meets its boundary orthogonally, Mn hypersurface of Rn+1 given by the inverse image of γ under the canonical projection onto Rn+1/G,

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 4 / 10

The hypersurface case: more general examples

cohomogeneity two compact subgroup G ⊂ SO(n + 1), γ curve that is either contained in the interior of Rn+1/G or meets its boundary orthogonally, Mn hypersurface of Rn+1 given by the inverse image of γ under the canonical projection onto Rn+1/G, Mn is a cohomogeneity one hypersurface, called the standard examples.

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 4 / 10

The hypersurface case: more general examples

cohomogeneity two compact subgroup G ⊂ SO(n + 1), γ curve that is either contained in the interior of Rn+1/G or meets its boundary orthogonally, Mn hypersurface of Rn+1 given by the inverse image of γ under the canonical projection onto Rn+1/G, Mn is a cohomogeneity one hypersurface, called the standard examples.

Theorem (Mercuri-Podestà-Seixas-Tojeiro, Comment. Math. Helv., 2006):

Let f : Mn → Rn+1 be a complete hypersurface of G-cohomogeneity one.

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 4 / 10

The hypersurface case: more general examples

cohomogeneity two compact subgroup G ⊂ SO(n + 1), γ curve that is either contained in the interior of Rn+1/G or meets its boundary orthogonally, Mn hypersurface of Rn+1 given by the inverse image of γ under the canonical projection onto Rn+1/G, Mn is a cohomogeneity one hypersurface, called the standard examples.

Theorem (Mercuri-Podestà-Seixas-Tojeiro, Comment. Math. Helv., 2006):

Let f : Mn → Rn+1 be a complete hypersurface of G-cohomogeneity one. Assume that n ≥ 3 and Mn is compact or that n ≥ 5 and the connected components of the flat part of Mn are bounded.

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 4 / 10

The hypersurface case: more general examples

cohomogeneity two compact subgroup G ⊂ SO(n + 1), γ curve that is either contained in the interior of Rn+1/G or meets its boundary orthogonally, Mn hypersurface of Rn+1 given by the inverse image of γ under the canonical projection onto Rn+1/G, Mn is a cohomogeneity one hypersurface, called the standard examples.

Theorem (Mercuri-Podestà-Seixas-Tojeiro, Comment. Math. Helv., 2006):

Let f : Mn → Rn+1 be a complete hypersurface of G-cohomogeneity one. Assume that n ≥ 3 and Mn is compact or that n ≥ 5 and the connected components of the flat part of Mn are bounded. Then f is either rigid or a rotational hypersurface.

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 4 / 10

How about isometric immersions f : Mn → Rn+k, with k ≥ 2?

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 5 / 10

How about isometric immersions f : Mn → Rn+k, with k ≥ 2?

Theorem (Castro-Noronha, Geom. Dedicata, 1999):

Let f : Mn → Rn+2, n ≥ 5, be an isometric immersion of a compact homogeneous Riemannian manifold.

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 5 / 10

How about isometric immersions f : Mn → Rn+k, with k ≥ 2?

Theorem (Castro-Noronha, Geom. Dedicata, 1999):

Let f : Mn → Rn+2, n ≥ 5, be an isometric immersion of a compact homogeneous Riemannian

• manifold. Then f is either a homogeneous isoparametric hypersurface of Sn+1, or isometric to Sn
• r is isometrically covered by R × Sn−1.

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 5 / 10

Cohomogeneity one Riemannian manifolds

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 6 / 10

Cohomogeneity one Riemannian manifolds

Mn Riemannian manifold acted on by a connected closed subgroup G ⊂ Iso(Mn),

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 6 / 10

Cohomogeneity one Riemannian manifolds

Mn Riemannian manifold acted on by a connected closed subgroup G ⊂ Iso(Mn), γ : R → M normal geodesic on M (it crosses each orbit orthogonally),

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 6 / 10

Cohomogeneity one Riemannian manifolds

Mn Riemannian manifold acted on by a connected closed subgroup G ⊂ Iso(Mn), γ : R → M normal geodesic on M (it crosses each orbit orthogonally), J = (a, b) ⊂ R open interval such that π ◦ γ is a homeomorphism of J onto π(Mr),

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 6 / 10

Cohomogeneity one Riemannian manifolds

Mn Riemannian manifold acted on by a connected closed subgroup G ⊂ Iso(Mn), γ : R → M normal geodesic on M (it crosses each orbit orthogonally), J = (a, b) ⊂ R open interval such that π ◦ γ is a homeomorphism of J onto π(Mr), For each t ∈ J denote by k(t) the principal curvature of the orbit G(γ(t)) with respect to the normal vector field X(g · γ(t)) = g∗γ′(t).

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 6 / 10

Cohomogeneity one Riemannian manifolds

Mn Riemannian manifold acted on by a connected closed subgroup G ⊂ Iso(Mn), γ : R → M normal geodesic on M (it crosses each orbit orthogonally), J = (a, b) ⊂ R open interval such that π ◦ γ is a homeomorphism of J onto π(Mr), For each t ∈ J denote by k(t) the principal curvature of the orbit G(γ(t)) with respect to the normal vector field X(g · γ(t)) = g∗γ′(t). For a fixed t0 ∈ J, define the function ρ by ρ(t) = e

− t

t0 k(s)ds. Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 6 / 10

Cohomogeneity one Riemannian manifolds

Mn Riemannian manifold acted on by a connected closed subgroup G ⊂ Iso(Mn), γ : R → M normal geodesic on M (it crosses each orbit orthogonally), J = (a, b) ⊂ R open interval such that π ◦ γ is a homeomorphism of J onto π(Mr), For each t ∈ J denote by k(t) the principal curvature of the orbit G(γ(t)) with respect to the normal vector field X(g · γ(t)) = g∗γ′(t). For a fixed t0 ∈ J, define the function ρ by ρ(t) = e

− t

t0 k(s)ds.

Set p0 = γ(t0) and consider the action of G on J ×ρ G(p0) given by g · (t, p) = (t, g(p)).

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 6 / 10

Cohomogeneity one Riemannian manifolds

Mn Riemannian manifold acted on by a connected closed subgroup G ⊂ Iso(Mn), γ : R → M normal geodesic on M (it crosses each orbit orthogonally), J = (a, b) ⊂ R open interval such that π ◦ γ is a homeomorphism of J onto π(Mr), For each t ∈ J denote by k(t) the principal curvature of the orbit G(γ(t)) with respect to the normal vector field X(g · γ(t)) = g∗γ′(t). For a fixed t0 ∈ J, define the function ρ by ρ(t) = e

− t

t0 k(s)ds.

Set p0 = γ(t0) and consider the action of G on J ×ρ G(p0) given by g · (t, p) = (t, g(p)).

Theorem (Podestà-Spiro 1995; Moutinho 2006):

Suppose that the principal orbits are umbilical hypersurfaces of Mn.

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 6 / 10

Cohomogeneity one Riemannian manifolds

Mn Riemannian manifold acted on by a connected closed subgroup G ⊂ Iso(Mn), γ : R → M normal geodesic on M (it crosses each orbit orthogonally), J = (a, b) ⊂ R open interval such that π ◦ γ is a homeomorphism of J onto π(Mr), For each t ∈ J denote by k(t) the principal curvature of the orbit G(γ(t)) with respect to the normal vector field X(g · γ(t)) = g∗γ′(t). For a fixed t0 ∈ J, define the function ρ by ρ(t) = e

− t

t0 k(s)ds.

Set p0 = γ(t0) and consider the action of G on J ×ρ G(p0) given by g · (t, p) = (t, g(p)).

Theorem (Podestà-Spiro 1995; Moutinho 2006):

Suppose that the principal orbits are umbilical hypersurfaces of Mn. Then the map ψ : J ×ρ G(p0) → Mr given by ψ(t, g(p0)) = g(γ(t)) is an equivariant isometry with respect to the actions of G on the spaces J ×ρ G(p0) and Mr.

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 6 / 10

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 7 / 10

Definition:

An isometric immersion f : Ll ×ρ Mm → Qn

c is said to be a warped product of isometric

immersions determined by a warped product representation φ : V n−k ×σ Qk

˜ c → Qn c, onto an open

dense subset of Qn

c, if there exist isometric immersions h1 : Ll → V n−k and h2 : Mm → Qk ˜ c such

that ρ = σ ◦ h1

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 7 / 10

Definition:

An isometric immersion f : Ll ×ρ Mm → Qn

c is said to be a warped product of isometric

immersions determined by a warped product representation φ : V n−k ×σ Qk

˜ c → Qn c, onto an open

dense subset of Qn

c, if there exist isometric immersions h1 : Ll → V n−k and h2 : Mm → Qk ˜ c such

that ρ = σ ◦ h1 and the following diagram commutes: V n−k×σ Qk

˜ c

Ll

h1

• ×ρMm

h2

• φ
• f=φ◦(h1×h2)

Qn

c

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 7 / 10

Definition:

An isometric immersion f : Ll ×ρ Mm → Qn

c is said to be a warped product of isometric

immersions determined by a warped product representation φ : V n−k ×σ Qk

˜ c → Qn c, onto an open

dense subset of Qn

c, if there exist isometric immersions h1 : Ll → V n−k and h2 : Mm → Qk ˜ c such

that ρ = σ ◦ h1 and the following diagram commutes: V n−k×σ Qk

˜ c

Ll

h1

• ×ρMm

h2

• φ
• f=φ◦(h1×h2)

Qn

c

Example 1:

If h2 is an isometry, then f is called a rotational submanifold with profile h1.

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 7 / 10

Definition:

An isometric immersion f : Ll ×ρ Mm → Qn

c is said to be a warped product of isometric

immersions determined by a warped product representation φ : V n−k ×σ Qk

˜ c → Qn c, onto an open

dense subset of Qn

c, if there exist isometric immersions h1 : Ll → V n−k and h2 : Mm → Qk ˜ c such

that ρ = σ ◦ h1 and the following diagram commutes: V n−k×σ Qk

˜ c

Ll

h1

• ×ρMm

h2

• φ
• f=φ◦(h1×h2)

Qn

c

Example 1:

If h2 is an isometry, then f is called a rotational submanifold with profile h1. Geometrically, this means that V n−k is a half-space of a totally geodesic submanifold Qn−k

c

⊂ Qn

c bounded by a

totally geodesic submanifold Qn−k−1

c

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 7 / 10

Definition:

An isometric immersion f : Ll ×ρ Mm → Qn

c is said to be a warped product of isometric

immersions determined by a warped product representation φ : V n−k ×σ Qk

˜ c → Qn c, onto an open

dense subset of Qn

c, if there exist isometric immersions h1 : Ll → V n−k and h2 : Mm → Qk ˜ c such

that ρ = σ ◦ h1 and the following diagram commutes: V n−k×σ Qk

˜ c

Ll

h1

• ×ρMm

h2

• φ
• f=φ◦(h1×h2)

Qn

c

Example 1:

If h2 is an isometry, then f is called a rotational submanifold with profile h1. Geometrically, this means that V n−k is a half-space of a totally geodesic submanifold Qn−k

c

⊂ Qn

c bounded by a

totally geodesic submanifold Qn−k−1

c

and f(Ll ×ρ Mm) is the submanifold of Qn

c generated by the

action on h1(L) of the subgroup of isometries of Qn

c that leave Qn−k−1 c

invariant.

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 7 / 10

Definition:

An isometric immersion f : Ll ×ρ Mm → Qn

c is said to be a warped product of isometric

immersions determined by a warped product representation φ : V n−k ×σ Qk

˜ c → Qn c, onto an open

dense subset of Qn

c, if there exist isometric immersions h1 : Ll → V n−k and h2 : Mm → Qk ˜ c such

that ρ = σ ◦ h1 and the following diagram commutes: V n−k×σ Qk

˜ c

Ll

h1

• ×ρMm

h2

• φ
• f=φ◦(h1×h2)

Qn

c

Example 1:

If h2 is an isometry, then f is called a rotational submanifold with profile h1. Geometrically, this means that V n−k is a half-space of a totally geodesic submanifold Qn−k

c

⊂ Qn

c bounded by a

totally geodesic submanifold Qn−k−1

c

and f(Ll ×ρ Mm) is the submanifold of Qn

c generated by the

action on h1(L) of the subgroup of isometries of Qn

c that leave Qn−k−1 c

invariant.

Example 2:

If h1 is a local isometry then, for c = 0, we have that f(Ll ×ρ Mm) is contained in the product of an Euclidean factor Rn−k−1 with a cone in Rk+1 over h2.

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 7 / 10

Main theorem

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 8 / 10

Main theorem

Theorem (–, Silva):

Let f : Mn → Rn+2, with n ≥ 4, be an isometric immersion of a compact Riemannian manifold of cohomogeneity one under the action of a closed Lie subgroup G of Iso(M).

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 8 / 10

Main theorem

Theorem (–, Silva):

Let f : Mn → Rn+2, with n ≥ 4, be an isometric immersion of a compact Riemannian manifold of cohomogeneity one under the action of a closed Lie subgroup G of Iso(M). If the principal orbits under the action of G are umbilic hypersurfaces in Mn then one of the following possibilities holds:

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 8 / 10

Main theorem

Theorem (–, Silva):

Let f : Mn → Rn+2, with n ≥ 4, be an isometric immersion of a compact Riemannian manifold of cohomogeneity one under the action of a closed Lie subgroup G of Iso(M). If the principal orbits under the action of G are umbilic hypersurfaces in Mn then one of the following possibilities holds: (i) There exist a compact homogeneous hypersurface h : Mn−1 → Sn

c, a unit speed curve

λ : J = (a, b) → R2

+ and an isometry ψ : J ×ρ Mn−1 → Mr such that f ◦ ψ is the warped

product of λ with h determined by a warped product representation Φ : R2

+ ×σ Sn c → Rn+2.

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 8 / 10

Main theorem

Theorem (–, Silva):

Let f : Mn → Rn+2, with n ≥ 4, be an isometric immersion of a compact Riemannian manifold of cohomogeneity one under the action of a closed Lie subgroup G of Iso(M). If the principal orbits under the action of G are umbilic hypersurfaces in Mn then one of the following possibilities holds: (i) There exist a compact homogeneous hypersurface h : Mn−1 → Sn

c, a unit speed curve

λ : J = (a, b) → R2

+ and an isometry ψ : J ×ρ Mn−1 → Mr such that f ◦ ψ is the warped

product of λ with h determined by a warped product representation Φ : R2

+ ×σ Sn c → Rn+2.

R2

+ ×σ

Sn

c

J

λ

• ×ρ Mn−1

h

• φ
• f◦ψ=φ◦(λ×h)

Rn+2

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 8 / 10

Main theorem

Theorem (–, Silva):

Let f : Mn → Rn+2, with n ≥ 4, be an isometric immersion of a compact Riemannian manifold of cohomogeneity one under the action of a closed Lie subgroup G of Iso(M). If the principal orbits under the action of G are umbilic hypersurfaces in Mn then one of the following possibilities holds: (ii) There exist a compact surface h : M2 → R4 of intrinsic cohomogeneity one under the action

• f S1 and an isometry ψ : M2 ×ρ Sn−2

c

→ Mr such that f ◦ ψ is the warped product of h with the identity map i : Sn−2

c

→ Sn−2

c

determined by a warped product representation Φ : R4 ×σ Sn−2

c

→ Rn+2.

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 9 / 10

Main theorem

Theorem (–, Silva):

Let f : Mn → Rn+2, with n ≥ 4, be an isometric immersion of a compact Riemannian manifold of cohomogeneity one under the action of a closed Lie subgroup G of Iso(M). If the principal orbits under the action of G are umbilic hypersurfaces in Mn then one of the following possibilities holds: (ii) There exist a compact surface h : M2 → R4 of intrinsic cohomogeneity one under the action

• f S1 and an isometry ψ : M2 ×ρ Sn−2

c

→ Mr such that f ◦ ψ is the warped product of h with the identity map i : Sn−2

c

→ Sn−2

c

determined by a warped product representation Φ : R4 ×σ Sn−2

c

→ Rn+2. R4 ×σ Sn−2

c

M2

h

• ×ρ Sn−2

c id

• φ
• f◦ψ=φ◦(h×id)

Rn+2

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 9 / 10

Main theorem

Theorem (–, Silva):

Let f : Mn → Rn+2, with n ≥ 4, be an isometric immersion of a compact Riemannian manifold of cohomogeneity one under the action of a closed Lie subgroup G of Iso(M). If the principal orbits under the action of G are umbilic hypersurfaces in Mn then one of the following possibilities holds: (iii) There exist a unit speed curve λ : J = (a, b) → R2

+ and an isometry ψ : J ×ρ Sn−1 c

→ Mr such that f ◦ ψ = F ◦ G, where G is the warped product of λ with the identity map id : Sn−1

c

→ Sn−1

c

determined by a warped product representation φ : R2

+ ×σ Sn−1 c

→ Rn+1, and F : W → Rn+2 is an isometric immersion of an open subset W ⊂ Rn+1 that contains G(J ×ρ Sn−1

c

).

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 10 / 10

Main theorem

Theorem (–, Silva):

Let f : Mn → Rn+2, with n ≥ 4, be an isometric immersion of a compact Riemannian manifold of cohomogeneity one under the action of a closed Lie subgroup G of Iso(M). If the principal orbits under the action of G are umbilic hypersurfaces in Mn then one of the following possibilities holds: (iii) There exist a unit speed curve λ : J = (a, b) → R2

+ and an isometry ψ : J ×ρ Sn−1 c

→ Mr such that f ◦ ψ = F ◦ G, where G is the warped product of λ with the identity map id : Sn−1

c

→ Sn−1

c

determined by a warped product representation φ : R2

+ ×σ Sn−1 c

→ Rn+1, and F : W → Rn+2 is an isometric immersion of an open subset W ⊂ Rn+1 that contains G(J ×ρ Sn−1

c

). R2

+ ×σ Sn−1 c

J

λ

• ×ρ Sn−1

c id

• φ

Rn+1

F

• f◦ψ=F◦ψ◦(λ×id) Rn+2

Fernando Manfio Complete Euclidean submanifolds Santiago de Compostela 10 / 10