Coding of hypersurfaces in Euclidean spaces by a constant vector

1 Introduction

In differential geometry, the most studied topic is the geometry of hypersurfaces in Euclidean space. There is a trend of studying biharmonic hypersurfaces and hypersurfaces of finite type in Euclidean space (cf. [1–4]). Moreover, there is an interesting class of hypersurfaces having two distinct principal curvatures in Euclidean space, and these hypersurfaces are classified in [5,6].

Given an n -dimensional Riemannian manifold ( N n , g ) that is isometrically immersed in Euclidean space R n + 1 , in this article, we observe that each unit constant vector field a → on Euclidean space R n + 1 watches the hypersurface ( N n , g ) through two quantities it assigns to the hypersurface ( N n , g ) : one is a vector field v that is tangential component of a → called the code vector, and the other is the function ρ that is the normal component of a → called the coding function. These two quantities, the code vector v and coding function ρ of the hypersurface ( N n , g ) with respect to the constant unit vector field a → on Euclidean space R n + 1 , play an important role in shaping the geometry of the hypersurface, which we wish to explore in this article.

Let ( N n , g ) be an n -dimensional Riemannian manifold isometrically immersed in Euclidean space R n + 1 with unit normal ζ and shape operator S . Throughout this article, we fix a constant unit vector field a → on Euclidean space R n + 1 and express its restriction to the hypersurface ( N n , g ) as follows:

As mentioned earlier, we call the vector field v in equation (1.1) as the code vector and ρ as the coding function of the hypersurface. Grouping these all together, we call the sextuple ( N n , g , ζ , S , v , ρ ) a coded hypersurface of Euclidean space R n + 1 with respect to the constant unit vector a → .

Consider the sphere ( S b n , g ) of constant curvature b as an embedded hypersurface of R n + 1 given by

Then the shape operator S and the unit normal ζ of ( S b n , g ) are given by S = − b I , and ζ = b φ , where φ : S b n → R n + 1 is φ ( u ) = u . The code vector v and the coding function ρ of ( S b n , g ) gives a coded hypersurface ( S b n , g , b φ , − b I , v , ρ ) , of the Euclidean space R n + 1 , whose code vector v and coding function ρ satisfy (owing to Gauss-Weingarten formulae):

where ∇ X is the directional derivative with respect to the metric g and ∇ ρ is the gradient of coding function ρ . The aforementioned equation conveys that for the coded hypersurface ( S b n , g , b φ , − b I , v , ρ ) , we have

where Hess ( ρ ) is the Hessian of the coding function ρ and Δ ρ is its Laplacian. As the Ricci tensor Ric and the scalar curvature τ of ( S b n , g ) are given by R i c = ( n − 1 ) b g , τ = n ( n − 1 ) b , on employing equation (1.2), we see that the coded hypersurface ( S b n , g , b φ , − b I , v , ρ ) satisfies the static perfect fluid equation (SPFE) (cf. [7])

Observe that on ( S b n , g ) , one has Ric ( v , v ) = ( n − 1 ) b ∥ v ∥ 2 and the mean curvature H = − b , and therefore,

Also, on ( S b n , g ) , by equation (1.2), one has ρ Δ ρ = − n b ρ 2 , and by integrating this equation while using equation (1.1), we have

By integrating equation (1.4) and using the aforementioned equation, with ( S b n , g ) , we obtain

The aforementioned equation holds for code vector v and the coding function ρ and the mean curvature H of the sphere ( S b n , g ) as hypersurface of Euclidean space R n + 1 . This raises a question: Is equation (1.5) a characteristic property of the sphere ( S b n , g ) ? In other words, is a compact and simply connected coded hypersurface ( N n , g , ζ , S , v , ρ ) with mean curvature H satisfying equation (1.5) necessarily isometric to ( S b n , g ) ? We answer this question in affirmative and show that a a compact and simply connected coded hypersurface ( N n , g , ζ , S , v , ρ ) of Euclidean space R n + 1 with mean curvature H satisfies

if and only if the mean curvature H is a constant and that ( N n , g , ζ , S , v , ρ ) is isometric to ( S b n , g ) with b = H 2 (Theorem 1).

In other consideration, we observe for the coded hypersurface ( S b n , g , b φ , − b I , v , ρ ) , the coding function ρ satisfies the SPFE, that is, equation (1.3). Moreover, the shape operator S = − b I is invariant under the code vector v , that is, it satisfies

where £ v is the Lie derivative with respect to code vector v . One observes that equations (1.3) and (1.6) are characteristics of the coded hypersurface ( S b n , g , b φ , − b I , v , ρ ) of the Euclidean space R n + 1 . This raises another question: Is a complete and simply connected coded hypersurface ( N n , g , ζ , S , v , ρ ) of Euclidean space with coding function ρ satisfying SPFE and shape operator S invariant under the code vector v isometric to a sphere? We show that the answer to aforementioned question is in affirmative provided the coded hypersurface is not minimal, that is, H ≠ 0 (Theorem 2).

Next, in this article, we show that a complete and simply connected coded hypersurface ( N n , g , ζ , S , v , ρ ) , having a point p ∈ N n , where Ric ( v , v ) ( p ) > 0 , the coding function ρ satisfying Δ ρ = − n c ρ for a positive constant c , mean curvature H satisfying H 2 ≥ c , and H is a constant along integral curves of the code vector v , necessarily implies that ( N n , g , ζ , S , v , ρ ) is isometric to the sphere S c n and the converse also holds (Theorem 3).

2 Preliminaries

Let ( N n , g , ζ , S , v , ρ ) a coded hypersurface of Euclidean space R n + 1 with respect to the constant unit vector a → . We denote by ∇ X the directional derivative with respect to the vector field X ∈ D ( N n ) corresponding to the Levi-Civita connection of the induced metric g on the coded hypersurface ( N n , g , ζ , S , v , ρ ) , where D ( N n ) is the set of all smooth vector fields on N n . Differentiating equation (1.1), while using Gauss and Weingarten formulas, and equating like parts, we obtain

where ∇ ρ is the gradient of the coding function. The mean curvature function H of ( N n , g , ζ , S , v , ρ ) is given by

where { E j } 1 n is a local frame on ( N n , g , ζ , S , v , ρ ) . The curvature tensor R , Ricci tensor Ric, and scalar curvature τ of ( N n , g , ζ , S , v , ρ ) are given by

and

where

The shape operator S of the coded hypersurface ( N n , g , ζ , S , v , ρ ) satisfies the Codazzi equation:

where ( ∇ S ) ( X 1 , X 2 ) = ∇ X 1 S X 2 − S ( ∇ X 1 X 2 ) . Combining equations (2.2) and (2.6), while using the symmetry of the operator S , it is easy to arrive at

For a smooth function φ on the coded hypersurface ( N n , g , ζ , S , v , ρ ) , the Hessian operator ℋ φ is defined by

and the Hessian Hess ( φ ) is given by

and the Laplacian Δ φ of φ is defined as Δ φ = div ( ∇ φ ) , which also satisfies

For more information on the concepts discussed earlier, we recommend consulting the books [8,9].

3 A coded hypersurface with restricted Ric ( v , v )

Let ( N n , g , ζ , S , v , ρ ) a coded hypersurface of the Euclidean space R n + 1 with respect to the constant unit vector a → and mean curvature H . In this section, we show that a proper lower bound on the integral of the Ricci curvature R ( v , v ) gives a characterization of a sphere ( S b n , g ) , as shown in the following result:

4 A coded hypersurface with shape operator invariant under code vector

Let ( N n , g , ζ , S , v , ρ ) be a coded hypersurface of Euclidean space R n + 1 with respect to the constant unit vector a → and mean curvature H and scalar curvature τ . We say that the shape operator S is invariant under the code vector v if

holds, where { f t } is the local flow of v , or equivalently

where £ v is the Lie differentiation in the direction of v . We also ask that the coding function ρ satisfies SPFE(static perfect fluid equation),

It is interesting to observe that the combination (4.1) and (4.2) for a complete and simply connected coded hypersurface ( N n , g , ζ , S , v , ρ ) of positive Ricci curvature gives a characterization of the sphere ( S b n , g ) as seen in the following result:

5 A coded hypersurface with coding function eigenvalue of Laplace operator

Let ( N n , g , ζ , S , v , ρ ) be a coded hypersurface of Euclidean space R n + 1 with respect to the constant unit vector a → of mean curvature H and scalar curvature τ . In this section, we seek the result of a restriction on the Laplacian of the coding function ρ . As we know the sphere ( S b n , g ) as a coded hypersurface ( S b n , g , b φ , − b I , v , ρ ) has coding function ρ satisfying Δ ρ = − n b ρ , and this motivates the question: Under what conditions a complete simply connected coded hypersurface ( N n , g , ζ , S , v , ρ ) of the Euclidean space R n + 1 with coding function ρ satisfying Δ ρ = − n b ρ , where b is a positive constant, is isometric to ( S b n , g ) ? We answer this question through the following:

6 Conclusions

In Theorem 2, we have studied a complete and simply connected coded hypersurface ( N n , g , ζ , S , v , ρ ) of Euclidean space R n + 1 , n > 1 , with positive Ricci curvature that has the shape operator S invariant under the code vector v and the coding function ρ satisfies SPFE, and obtained a characterization of the sphere ( S b n , g ) . The scope of studying geometry of coded hypersurface ( N n , g , ζ , S , v , ρ ) in Euclidean space R n + 1 is quite modest. For instance, it will be interesting to study the impact of the restrictions on the code vector v , such as v is a conformal vector field or v is a geodesic vector field (cf. [11,12]). Equation (1.1) suggests that in the case, the code vector v of a compact and simply connected coded hypersurface ( N n , g , ζ , S , v , ρ ) of Euclidean space R n + 1 being conformal vector field with suitable further restrictions could lead to a characterization of the sphere ( S b n , g ) . However, in the case the code vector v of a complete and connected coded hypersurface ( N n , g , ζ , S , v , ρ ) of Euclidean space R n + 1 being geodesic vector field (i.e., integral curves of v are geodesics), through equation (2.2), under the assumption that coding function ρ ≠ 0 , one concludes coding function ρ is a constant and that S v = 0 . Thus, an interesting question would be to find suitable conditions under, which a complete and connected coded hypersurface ( N n , g , ζ , S , v , ρ ) of Euclidean space R n + 1 with code vector v a geodesic vector field and coding function ρ ≠ 0 is R × M n − 1 , where M n − 1 is a codimension 2 submanifold of R n + 1 (cf. [8]).