Hypersurfaces with a canonical principal direction

Garnica, Eugenio; Palmas, Oscar; Ruiz-Hernández, Gabriel

doi:10.1016/j.difgeo.2012.06.001

Cited by 23 publications

(25 citation statements)

References 14 publications

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“…Namely, if

Z^{⊤}

is a geodesic vector field of M then M has canonical principal direction with respect to Z (cf. [, Theorem 5, p. 385]).…”

Section: Preliminariesmentioning

confidence: 99%

“…Then we need to show that

Δ f = b \circ f

. The formula for the mean curvature

H_{f}

[, p. 106] of the level sets of f is

H_{f} = \frac{1}{| \nabla f |} false(- normalΔ f + \frac{1}{| \nabla f |} false⟨ \nabla f, \nabla false| \nabla f false| false⟩ false) .

Then

\begin{matrix} true \\ right H_{f} & = & left \frac{1}{false| \nabla f false|} (- Δ f + \frac{1}{| \nabla f |} false⟨ \nabla f, \nabla (a \circ f) false⟩) \\ = & left \frac{1}{false| \nabla f false|} (- Δ f + \frac{1}{| \nabla f |} false⟨ \nabla f, (a^{'} \circ f) \nabla f false⟩) \\ = & left \frac{- normalΔ f}{a \circ f} + a^{'} \circ f . \end{matrix}

Now by [, Proposition 13, p. 389] we get that

H_{f}

is constant i.e.

H_{f} = c \circ f

.…”

Section: Hypersurfaces With Canonical Principal Direction and Constanmentioning

confidence: 99%

“…Classical examples of CPD surfaces of the Euclidean space

R^{3}

are the so‐called molding surfaces given by the parametric patch:

S false(u, v false) : = M false(u false) + f false(v false) \vec{N} false(u false) + g false(v false) \vec{k}

where

M (u)

is a curve in the

x y

‐plane,

\vec{N} false(u false)

a unit normal vector at

M (u)

in the

x y

‐plane,

f (v), g (v)

two smooth functions and

\overset{k}{true}

the unit vector of the z ‐axis (see [, Remark 12, p. 388]). It is not difficult to see that the curves

v \to S (u_{0}, v)

are curvature lines hence

S (u, v)

is a CPD surface with respect to the constant vector field

\overset{k}{true}

.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

CMC hypersurfaces with canonical principal direction in space forms

Scala

Ruiz-Hernández

2016

Mathematische Nachrichten

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A hypersurface M⊂M¯ of the space form M¯ has a canonical principal direction (CPD) relative to the closed and conformal vector field Z of M¯ if the projection Z⊤ of Z to M is a principal direction of M. We show that CPD hypersurfaces with constant mean curvature are foliated by isoparametric hypersurfaces. In particular, we show that a CPD surface with constant mean curvature of space form M¯ is invariant by the flow of a Killing vector field whose action is polar on M¯. As consequence we show that a compact CPD minimal surface of the sphere S3 is a Clifford torus. Finally, we consider the case when a CPD Euclidean hypersurface has zero Gauss–Kronecker curvature.

show abstract

“…Namely, if

Z^{⊤}

is a geodesic vector field of M then M has canonical principal direction with respect to Z (cf. [, Theorem 5, p. 385]).…”

Section: Preliminariesmentioning

confidence: 99%

“…Then we need to show that

Δ f = b \circ f

. The formula for the mean curvature

H_{f}

[, p. 106] of the level sets of f is

H_{f} = \frac{1}{| \nabla f |} false(- normalΔ f + \frac{1}{| \nabla f |} false⟨ \nabla f, \nabla false| \nabla f false| false⟩ false) .

Then

\begin{matrix} true \\ right H_{f} & = & left \frac{1}{false| \nabla f false|} (- Δ f + \frac{1}{| \nabla f |} false⟨ \nabla f, \nabla (a \circ f) false⟩) \\ = & left \frac{1}{false| \nabla f false|} (- Δ f + \frac{1}{| \nabla f |} false⟨ \nabla f, (a^{'} \circ f) \nabla f false⟩) \\ = & left \frac{- normalΔ f}{a \circ f} + a^{'} \circ f . \end{matrix}

Now by [, Proposition 13, p. 389] we get that

H_{f}

is constant i.e.

H_{f} = c \circ f

.…”

Section: Hypersurfaces With Canonical Principal Direction and Constanmentioning

confidence: 99%

“…Classical examples of CPD surfaces of the Euclidean space

R^{3}

are the so‐called molding surfaces given by the parametric patch:

S false(u, v false) : = M false(u false) + f false(v false) \vec{N} false(u false) + g false(v false) \vec{k}

where

M (u)

is a curve in the

x y

‐plane,

\vec{N} false(u false)

a unit normal vector at

M (u)

in the

x y

‐plane,

f (v), g (v)

two smooth functions and

\overset{k}{true}

the unit vector of the z ‐axis (see [, Remark 12, p. 388]). It is not difficult to see that the curves

v \to S (u_{0}, v)

are curvature lines hence

S (u, v)

is a CPD surface with respect to the constant vector field

\overset{k}{true}

.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

CMC hypersurfaces with canonical principal direction in space forms

Scala

Ruiz-Hernández

2016

Mathematische Nachrichten

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show abstract

“…M is said to have a canonical principal direction relative to X if the tangential projection of X to M gives a principal direction. For example, a rotational hypersurface in Euclidean spaces has a canonical principal direction relative to a vector field parallel to its rotation axis, [12]. It turns out that whenM is a product spaceM × R or a semi-Euclidean space, some common interesting geometrical properties of hypersurfaces endowed with a canonical principal direction relative to X occur if X is chosen to be a fixed direction k (See Theorem 3.6, Theorem 3.…”

Section: Introductionmentioning

confidence: 99%

A Short Survey on Surfaces Endowed with a Canonical Principal Direction

Ergüt¹,

Kelleci²

2017

Proceedings Book of International Workshop on Theory of Submanifolds

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“…A natural extension of constant angle surfaces is the class of surfaces with a canonical principal direction, see the definitions in [4,7]. In recent years, there are many classification results in different ambient spaces concerning the constant angle surfaces and surfaces with canonical principal directions, for instance, see [5,6,9,12,16,17,19].…”

Section: Introductionmentioning

confidence: 99%

On Lorentz GCR Surfaces in Minkowski 3-Space

Fu¹,

Yang²

2016

Bulletin of the Korean Mathematical Society

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Abstract. A generalized constant ratio surface (GCR surface) is defined by the property that the tangential component of the position vector is a principal direction at each point on the surface, see [8] for details. In this paper, by solving some differential equations, a complete classification of Lorentz GCR surfaces in the three-dimensional Minkowski space is presented. Moreover, it turns out that a flat Lorentz GCR surface is an open part of a cylinder, apart from a plane and a CMC Lorentz GCR surface is a surface of revolution.

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Hypersurfaces with a canonical principal direction

Cited by 23 publications

References 14 publications

CMC hypersurfaces with canonical principal direction in space forms

CMC hypersurfaces with canonical principal direction in space forms

A Short Survey on Surfaces Endowed with a Canonical Principal Direction

On Lorentz GCR Surfaces in Minkowski 3-Space

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