Discovering Geometric and Topological Properties of Ellipsoids by Curvatures

Aims/ Objectives: We are interested in discovering the geometric, topological and physical properties of ellipsoids by analyzing curvature properties on ellipsoids. We begin with studying ellipsoids as a starting point. Our aim is to find a way to study geometric, topological and physical properties from the analytic curvature properties for convex hyper-surfaces in the general setting. Study Design: Multiple-discipline study between Differential Geometry, Topology and Mathematical Physics. Place and Duration of Study: Department of Mathematics (Borough of Manhattan Community College-The City University of New York), Department of Mathematics (University of Oklahoma), Department of Mathematics and Statistics (University of West Florida), and Department of Mathematics (Central Michigan University), between January 2014 and February 2015. *Corresponding author: E-mail: lwu@bmcc.cuny.edu Wu et al.; BJMCS, 8(4), 318-329, 2015; Article no.BJMCS.2015.165 Methodology: Calculating curvatures of a surface is now at the threshold of a better understanding regarding geometric, topological and physical properties on a surface. In order to calculate various curvatures, we demonstrate the way to compute the second fundamental form associated with curvatures by extending the calculation method from spheres to ellipsoids. Results: Just as curvatures of a sphere are determined by its radius, curvatures of an ellipsoid are determined by its longest axis and its shortest axis. On an ellipsoid, the value of the ratio of its longest axis to its shortest axis is also a critical index to characterize its geometric, topological and physical behaviors. Conclusion: Our results on ellipsoids are extensions or generalizations of results of LawsonSimons, Wei, and Simons on spheres, and Kobayashi-Ohnita-Takeuchi on an ellipsoid with “one variable”. Methods and research findings in this paper can lead to future research on convex hyper-surfaces.


Introduction
For a space curve, the curvature is an amount of the bending degree to measure on how sharply a curve bends. A smaller circle bends more sharply with the bigger value of the curvature due to the fact the curvature of a circle is equal to the reciprocal of its radius. For a smooth surface, the curvature is an index to visualize the degree of the deviation from being flat. Calculating curvatures of surfaces is now at the threshold of a better understanding regarding geometric, topological, and physical properties for various shapes of surfaces.
For a convex hyper-surface in a Riemannian manifold, studying geometric, topological, and physical properties by analyzing its curvature properties is an impressive idea. It has a lasting and profound impact on mathematicians. In the direction of geometric properties such as stability or instability reflected by curvatures associated with the second fundamental form, Wei and Yau [1] defined a p-super-strongly unstable (p-SSU) manifold. Yang-Mills [2] defined a Yang-Mills instable manifold. Howard and Wei [3] studied the stability or instability of sub-manifolds of a surface. Wei, Wu and Zhang [4] proved the instability of minimal immersion on a manifold with non-positive scalar curvature and generalized Bernstein-type theorem. In the direction of topological properties reflected by curvatures, Synge [5] found the link between the topological vanishing properties of the first homotopy classes and the properties of the sectional curvature. Wei and Wu [6] proved the relationship between the topological vanishing properties of the higher-dimensional homotopy classes and the properties of the Ricci curvature. In the direction of the physical shape of a surface reflected by curvatures, Wente [7] found a non-spherical finite surface as a counter-example of a Hopf's conjecture. This counter-example is a constant mean curvature surface with the shape of a self-intersecting threelobed torus.
In the study of a convex hyper-surface, most mathematicians began with a sphere as a starting point and have obtained numerous results on a sphere. For example, Wei [8] has obtained the topological vanishing theorem for minimal sub-manifolds of a sphere with the Ricci curvature bounded below. Lawson and Simons [9] have obtained the non-existence of stable sub-manifolds on a sphere.
In our paper, we are interested in an ellipsoid with (n + 1) variables where a sphere is a special case of an ellipsoid. We discover the topological vanishing properties and geometric properties of stability or instability on an ellipsoid, which are characterized by its curvature properties. The curvature values on an ellipsoid are determined by its longest axis and its shortest axis. The ratio of its longest axis to its shortest axis is considered as an indicator of an ellipsoid as a p-SSU manifold and a Yang-Mills instable manifold. In order to estimate curvature values, we demonstrate the way to compute the second fundamental form by extending the calculation method from a sphere to an ellipsoid. Our results on an ellipsoid are extensions or generalizations of results of Lawson and Simons [9], Wei [8], and Simons (announced in Tokyo in September of 1977, cf. [10], [11]) on spheres, and Kobayashi, Ohnita and Takeuchi [12] on an ellipsoid with "one variable".

Materials and Methods
Here and throughout this paper, we denote an ellipsoid by Note that an ellipsoid E n defined in our notation contains a non-spherical ellipsoid (i.e. ∃i = j, ai = aj) and a sphere (i.e. ai = r > 0, ∀1 ≤ i ≤ n + 1). We denote a sphere of radius r by We also denote a unit sphere by S n , i.e. r = 1. Let M n be a compact Riemannian manifold with possibly non-empty boundary, and let N k be isometrically immersed in R q . L p 1 (M n , N k ) denotes the set of maps u : M n → N k ( → R q ) whose component functions have weak first derivatives in L p and u(x) ∈ N k a.e. on M n .
where du denotes the differential of u, and dv is the volume element of M n .

Definition 2.2.
A map u ∈ L p 1 (M n , N k ) is said to be weakly p-harmonic (p > 1) if it is a weak solution to the following Euler-Lagrange equation for Ep on L p 1 (M n , N k ): div(|∇u| p−2 ∇u) = 0 u is called p-stable (resp. p-minimizing) if u is a local (resp. global) minimum of p-energy functional Ep within a homotopy class of L p 1 (M n , N k ) having the same trace on ∂M n .
Let's recall the definition of a p-strongly unstable (p-SU) manifold and the definition of a p-superstrongly unstable (p-SSU) manifold.

Definition 2.3.
A Riemannian manifold is p-strongly unstable (p-SU) if it is neither the domain nor the target of any non-constant smooth p-stable map.

Definition 2.4.
A Riemannian manifold N k with a Riemannian metric , N k is said to be p-superstrongly unstable (p-SSU) for p ≥ 2, if there exists an isometric immersion in R q such that, for every unit tangent vector X to N k at every point x ∈ N k , the following function is always negative: h is the second fundamental form of N k in R q , and {α1, · · · , α k } is a local orthonormal frame on N k near x.

Definition 2.5.
A p-SSU index ωp on a p-SSU manifold N k is defined to be: where φp(x) = inf X =1 (−Fp,x(X)) for every unit tangent vector X to N k at x and S(x) > 0 is a positive upper bound of the sectional curvature of N k at x.
Here are the definitions of a Yang-Mills functional of connections, a Yang-Mills weakly stable connection, and a Yang-Mills instable manifold. Definition 2.6. Let M n be a compact Riemannian manifold and P be a principal G-bundle over M n , where G is a compact Lie group. On the space CP of the connections in P , the Yang-Mills functional J : CP → R is defined by where Ω is the curvature of the connection ω in CP and where the norm is defined in terms of the Riemannian metric on M and a fixed AdG-invariant scalar product on the Lie algebra G of G.

Definition 2.7.
A critical point of J is called a Yang-Mills connection and its curvature a Yang-Mills field. A Yang-Mills connection ω is said to be weakly stable if the second variation of J at ω is nonnegative, i.e., d 2 dt 2 J(ωt)|t=0 ≥ 0 for every smooth family of connections ωt, −δ < t < δ, with ω0 = ω. We say that a compact Riemannian manifold M n is Yang-Mills instable if, for every choice of G and every principal G-bundle P over M n , none of the non-flat Yang-Mills connections in P is weakly stable.
We are interested in exploring geometric, topological and physical properties of ellipsoids by analyzing their curvature properties. Therefore, calculating curvatures on ellipsoids is the first crucial step. Furthermore, we find that calculating the second fundamental form plays an important role in calculating curvatures. The difficulty of calculating curvatures comes from the difficulty of calculating the second fundamental form. In our paper, we demonstrate the way to compute the second fundamental form by extending the calculation method from spheres S n (r) to ellipsoids E n .  [13]) Suppose ν is a unit normal vector field on an ellipsoid E n imbedded in R n+1 and X is any unit tangent vector on E n at a point x, then the second fundamental form hXX defined by where∇ is the Riemannian connection on R n+1 and , is the Riemannian metric on R n+1 .
Proof. Suppose {U1, U2, · · · , Un+1} is a standard orthonormal basis in R n+1 . Since E n is a level set of Ui and a unit normal vector (2.4)

Results and Discussion
We are interested in studying ellipsoids as one step to move on to the study of any convex hypersurface. Based on our calculation method for the second fundamental form (cf. Lemma 2.1) and curvatures (cf. Lemma 2.2) in the section 2, we will explore geometric, topological and physical properties on ellipsoids by estimating their curvature properties. Subjects on topological properties of vanishing theorems on ellipsoids, geometric properties of instability on ellipsoids, and applications of ellipsoids as different types of manifolds will be discussed in the following subsections.

Topological Vanishing Theorem on Minimal Sub-Manifolds of Ellipsoids
In this subsection, we study the topological vanishing properties on ellipsoids.
Theorem 3.1. (Topological Vanishing Theorem on Minimal Sub-Manifolds of Ellipsoids) Let N k be a minimal k-dimensional sub-manifold of an ellipsoid E n such that the Ricci curvature Ric N k of N k satisfies: Proof. At any point x ∈ N k , for any unit vector X ∈ TxN k , we choose an orthonormal frame {α1, · · · , α k } on N k such that X = α k . Let {β1, · · · , β n−k } be an orthonormal frame in the normal space (TxN k ) ⊥ on E n and ν be a unit normal vector to E n on R n+1 defined in Lemma 2.1. For 1 ≤ i, j ≤ k, let Rij (resp.Rij) denote the sectional curvature of N k (resp. an ellipsoid E n ) for the section αi ∧ αj. For 1 ≤ i, j ≤ k, lethij = ( α i αj) ⊥ where is the Riemannian connection on E n and () ⊥ is the projection onto the normal space (TxN k ) ⊥ . The fact that N k is minimal says: Let h(Y, Z) = ¯ Y ν, Z where¯ is the Riemannian connection on R n+1 , Y and Z are local vector fields on E n , and ν is a unit normal vector to E n . And the second fundamental form h of N k in R n+1 splits into the sum of the second fundamental formh of N k in E n and the second fundamental form h of E n in R n+1 . That is, Applying the Gauss-curvature equation, we have: Summing (3.3) over 1 ≤ i ≤ k and via (3.1), we have: Via (3.4) and the assumption Ric N k > k(1 − 1 p ) (max(a i )) 2 (min(a i )) 4 , we obtain: where we apply Lemma 2.2 forRij = Sec E n ≤ (max(a i )) 2 (min(a i )) 4 . On the other hand, by the formula of Q N k x (X), X N k for a minimal sub-manifold N k of E n (cf. [3] p.322), we have: Therefore, via (3.2), (3.5), (3.6) and applying Lemma 2.1 for hii(= hα i α i ) and h kk (= hα k α k = hXX ), we can get: Remark 3.1. Applying our results of Theorem 3.1 to unit spheres S n , we recapture Wei's results [8] to obtain the lower bound of Ricci curvature on unit spheres.

Non-existence of Stable Sub-Manifolds on Ellipsoids
In this subsection, by applying Trace Formulas, we discover the geometric properties of stability or instability on E n . [16]) Let R k (Ñ n , G) be the group of rectifiable currents of degree k iñ N n over the finitely generated abelian group G . If S ∈ R k (Ñ n , G) and x is a point at which S has an approximate tangent space Tx(S) then set − → Sx = e1 ∧ · · · ∧ e k where {e1, · · · , e k } is an orthonormal basis of Tx(S). Define IS to be the function on R n+m given by

Lemma 3.2. (Trace Formulas
where M is the mass and ϕ V T t is the flow (or one parameter pseudogroup) generated by V T . Then (where S is a variational measure associated with S) and if S is stable then trace(IS ) ≥ 0. If {e1, · · · , en+m} is an orthonormal basis of R n+m such that {e1, · · · , en} is an orthonormal basis of TÑ n x , {en+1, · · · , en+m} is an orthonormal basis of the normal space of TÑ n x in R n+m , and ξ = e1 ∧ · · · ∧ e k , then And if for some k, trace(Q ξ ) < 0 holds for all ξ = e1 ∧ · · · ∧ e k tangent toÑ n , then there are no stable currents of degree k in R k (Ñ , G) for any finitely generated abelian group G. In particular there are no closed stable minimal sub-manifolds of dimension k or n − k inÑ n . . In particular, there are no closed stable kdimensional sub-manifolds in a unit sphere S n for any 1 ≤ k < n.
Proof. Let N k be a k-dimensional sub-manifold of E n . Suppose {e1, · · · , e k } is an orthonormal basis in T N k x and {e1, · · · , en} is an orthonormal basis in T E n x . Then {e1, · · · , en, ν(= Z Z )} is an orthonormal basis in R n+1 where Z and ν are defined in Lemma 2.1. For any fixed i (1 ≤ i ≤ k), we have Applying Lemma 2.1 for hXX = he i e i = he r er , ∇ e i Z and Z , Lemma 3.2, and (3.9) we get In particular, trace(Q ξ ) < 0 on S n for any 1 ≤ k < n.
Remark 3.4. Our results of Theorem 3.3 extend Lawson and Simons' result [9] from spheres to ellipsoids.

Application of Ellipsoids as p-Super-Strongly Unstable (p-SSU) Manifolds
In this subsection, we discuss the criteria for ellipsoids as p-SSU manifolds. Furthermore, we calculate p-SSU index ωp when ellipsoids are p-SSU manifolds.
In particular, ωp = n−p n+p−2 on a unit sphere S n . Proof.
Remark 3.5. Wei and Yau [1] calculated p-SSU index ωp on p-SSU spheres. We extend their results of p-SSU index ωp to p-SSU ellipsoids.

Application of Ellipsoids as Yang-Mills Instable Manifolds
In this subsection, we find the criteria for ellipsoids as Yang-Mills instable manifolds.
Theorem 3.6. (Criteria for Ellipsoids as Yang-Mills Instable Manifolds) An ellipsoid E n is Yang-Mills instable if µ 3 < n−2 2 . In particular, a unit n-sphere S n is Yang-Mills instable for n > 4.
On a convex hyper-surface, how can we link the curvatures properties to the geometric, topological, and physical properties? Answering this question will be very challenging. Methods and research findings from the point of view on ellipsoids in this paper provide a clue to the future research on convex hyper-surfaces.