Some Theorems on Austere Submanifolds

Bogdan Suceavă

Abstract

In §1 there are described the introductory notions of the geometry of submanifolds, as they are required in the paper. There are introduced the austere submanifold and are given same examples. The shape discriminant, depending upon the principal curvatures of the submanifold, is introduced in §2, repeating the idea from [8]. Here are proved completely all its properties necessary to prepare the main results, which are stated in §3, where the geometry of the austere submanifolds is presented related on some local techniques involving the principal curvatures.

Mathematics Subject Classification: 53B25, 53B35, 53C42 Key words: real submanifold, austere submanifold, Kaehler manifold, totally geodesic submanifold

1 Preliminaries

Let M be an n-dimensional Riemannian manifold isometrically imersed in an almost complex Hermitian n + m-dimensional manifold M̄ with almost complex structure J and almost Hermitian metric g. For any vector X tangent to M we write

(1.1)  JX = P X + F X,

where P X and F X are the tangent and the normal components of JX, respectively. For any point p ∈ M we have the Gauss and Weingarten formulas

(1.2)  ( ¯∇_{X}Y)_{p} = (∇_{X}Y)_{p} + h_{p}(X, Y), ∀X, Y ∈ T M

(1.3)  ( ¯∇_{X}ξ)_{p} = −(A_{ξ}X)_{p} + (D_{X}ξ)_{p}, ∀X ∈ T M, ∀ξ ∈ T^{⊥}M,

where ¯∇ and ∇ are the Levi-Civita connections in the Riemannian manifolds M̄ and M, h is the second fundamental form of the submanifold M of M̄, A_{ξ} the shape operator for the normal field ξ, and D the normal connection in T^{⊥}M.

We will denote (A_{ξ})_{p} not only the shape operator in p, but also the matrix associated to the linear mapping from T_{p}M to T_{p}M.

Balkan Journal of Geometry and Its Applications, Vol.2, No.1, 1997, pp. 109-115 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

Definition 1.1 The principal curvatures of the submanifold M associated to the field ξ ∈ T^{⊥}M are the roots of the characteristic polynomial

(1.4)  det((A_{ξ})_{p} − ρ(p)I_{n}) = 0.

Notation 1.1. Let ξ_{1}, ..., ξ_{m} be an orthonormal base for a neighbourhood U of the point p ∈ M, where ξ_{i}(i = \overline{1, m}) are sections in the normal bundle T^{⊥}M. For these normal fields ξ_{1}, ...ξ_{m} we have the principal curvatures, respectively:

ρ_{1}^{1}(p), ..., ρ_{n}^{1}(p)
.....................
ρ_{1}^{m}(p), ..., ρ_{n}^{m}(p)

Remark 1.1. Some properties of the principal curvatures are presented in [1], pg.67. Definition 1.2. Let M be a submanifold of the Riemannian manifold (M̄, g). M is called minimal if trace H = 0. M is called austere if for every ξ ∈ T^{⊥}M the family of the eigenvalues of A_{ξ} - repeated as many times as it is their multiplicity - is invariable by its multiplication with (-1). Remark 1.2. Obviously, an austere submanifold is minimal. Example 1.1.. The trivial case of austere submanifolds are the totally geodesic submanifolds. For example, for any hyperplane (H) ⊆ E^{n} = (R^{n}, can), all the eigenvalues are 0, and (H) is austere since (H) is totally geodesic. Example 1.2. The Clifford torus is an austere submanifold. Let f : R^{2} → R^{4} given by

f(θ, φ) = \frac{1}{\sqrt{2}}(cos θ, sin θ, cos φ, sin φ), (θ, φ) ∈ R^{2}.

The mapping f is an imersion of R^{2} into the unit sphere S^{3}(1) ⊂ R^{4}, where image f(R^{2}) is the flat torus. The vectors

e_{1} = (− sin θ, cos θ, 0, 0)
e_{2} = (0, 0, − sin φ, cos φ)

form an orthonormal basis of the tangent space T_{f(θ,φ)} Imf, and the normal vectors

η_{1} = \frac{1}{\sqrt{2}}(cos θ, sin θ, cos φ, sin φ)

η_{2} = \frac{1}{\sqrt{2}}(− cos θ, − sin θ, cos φ, sin φ)

form an orthonormal basis of the normal space. The matrices A_{η_{1}} and A_{η_{2}} with respect to the basis {e_{1}, e_{2}} are

A_{η_{1}} = ( \begin{matrix}−\frac{1}{\sqrt{2}} & 0 \\ 0 & −\frac{1}{\sqrt{2}} \end{matrix} ) ,

A_{η_{2}} = ( \begin{matrix}\frac{1}{\sqrt{2}} & 0 \\ 0 & −\frac{1}{\sqrt{2}} \end{matrix} ) .

As we can see, T_{f(θ,φ)}Imf ⊕ Rη_{2} = T_{f(θ,φ)}S^{3}(1) and T_{f(θ,φ)}S^{3}(1) ⊕ Rη_{1} =

T_{f(θ,φ)}E^{4}. The eigenvalues \frac{1}{\sqrt{2}} and −\frac{1}{\sqrt{2}} show us that the Clifford torus is an austere

submanifold of the unit sphere S^{3}.

2 The shape discriminant

We follow now the construction from [8]. We begin with the following Lemma 2.1. Let g ∈ R[X] be the polynomial

g(X) = a_{0}X^{n} + a_{1}X^{n−1} + a_{2}X^{n−2} + ... + a_{n}

with a_{0} ≠ 0 and n > 1. If all the roots of g are real, then

Δ(g, n) = \frac{2(n − 1)}{n}a_{1}^{2} − 4a_{0}a_{2} ≥ 0.

Proof. If g has all the roots real, then g′ has also all the roots real. After (n − 2) steps, g^{(n−2)} has all the roots real. Actually, g^{(n−2)} is a polynomial of degree 2 and the condition x_{1}, x_{2} ∈ R means Δ(g, n) ≥ 0. Remark 2.1. The converse of this lemma is not true. Lemma 2.2. Let x_{i} ∈ R, i = \overline{1, n}. If ∑_{i=1}^{n} x_{i} = 0, then we have

(2.1.)  ( max_{i=\overline{1,n}} x_{i} − min_{i=\overline{1,n}} x_{i})^{2} ≤ −4 ∑_{1≤i<j≤n} x_{i}x_{j}.

Proof. If we suppose x_{1} ≤ x_{2} ≤ ... ≤ x_{n} we have:

( max_{i=\overline{1,n}} x_{i} − min_{i=\overline{1,n}} x_{i})^{2} = (x_{n} − x_{1})^{2}.

The condition ∑_{i=1}^{n} x_{i} = 0 gives

∑_{i=1}^{n} x_{i}^{2} + 2 ∑_{1≤i<j≤n} x_{i}x_{j} = 0,

and to prove the claim we have to show that

(x_{n} − x_{1})^{2} ≤ 2 ∑_{i=1}^{n} x_{i}^{2}

or 0 ≤ (x_{1} + x_{n})^{2} + 2(x_{2}^{2} + x_{3}^{2} + ... + x_{n−1}^{2}), q.e.d. Definition 2.1. Let M be a submanifold of the Riemannian manifold (M̄,g). The mapping

(2.2)  δ_{ξ} : M → R, δ_{ξ}(p) = Δ(det((A_{ξ})_{p} − ρ(p)I_{n}), n)

is called the shape discriminant of the submanifold M associated to the normal vector field ξ ∈ T^{⊥}M. Remark 2.2. Because the matrix (A_{ξ})_{p} is symmetric and its eigenvalues are all real, the lemma 2.1 gives

(2.3)  δ_{ξ}(p) ≥ 0, ∀p ∈ M, ∀ξ ∈ T^{⊥}M.

Notation 2.1. Let ξ_{1}, ...ξ_{m} be m normal sections in the normal bundle T^{⊥}M. We denote

δ_{i} = δ_{ξ_{i}} ; A_{i} = A_{ξ_{i}}.

For every section ξ_{i} we have the set of eigenvalues ρ_{1}^{i}, ..., ρ_{n}^{i}. Let L_{i} : M → R be the mapping which measures the length of the interval where lies the eigenvalues

(2.4)  L_{i}(p) = max_{j=\overline{1,n}} ρ_{j}^{i}(p) − min_{j=\overline{1,n}} ρ_{j}^{i}(p).

Remark 2.3. With the notation

||A_{i}||^{2} = (ρ_{1}^{i})^{2} + ... + (ρ_{n}^{i})^{2}

the shape discriminant is

(2.5)  δ_{i} = 2||A_{i}||^{2} − \frac{2}{n}(trA_{i})^{2}.

Theorem 2.1. With the notations above, for any p ∈ M and any basis ξ_{1}, ..., ξ_{m} in T^{⊥}M, we have the double inequality

(2.6)  \sqrt{( \begin{matrix}n \\ 2 \end{matrix} )^{−1} δ_{i}(p)} ≤ L_{i}(p) ≤ \sqrt{δ_{i}(p)}, ∀i = \overline{1, n}.

Proof. Let f ∈ R[X] the polynomial corresponding to the characteristic equation of the shape operator A_{i} in p ∈ M,

f(X) = a_{0}X^{n} + a_{1}X^{n−1} + ... + a_{n}.

We have seen that f^{(k)}(X) = 0 is an equation with all the roots real, ∀k = \overline{1, n − 1}, and for every k = \overline{1, n − 1}, we have

min_{i=\overline{1,n}} x_{i} ≤ min_{i=\overline{1,n}} x_{i}^{(k)} ≤ max_{j=\overline{1,n}} x_{j}^{(k)} ≤ max_{j=\overline{1,n}} x_{j}.

For k = n − 2, we find

|x_{1}^{n−2)} − x_{2}^{(n−2)}| = \frac{1}{|a_{0}|\sqrt{\frac{n(n−1)}{2}}} \sqrt{Δ}_{i},

and |a_{0}| = 1 proves the first inequality. To prove the second one, we use the lemma

2.2. We denote y_{i} = x_{i} + \frac{a_{1}}{na_{0}} , and ∑_{i=1}^{n} y_{i} = ∑_{i=1}^{n} x_{i} + ∑_{i=1}^{n} \frac{a_{1}}{na_{0}} = 0, from the Viète’s

formula.

For y_{i} chosen as above, we can apply the lemma 2.2:

( max_{i=\overline{1,n}} y_{i} − min_{i=\overline{1,n}} y_{i})^{2} ≤ −4 ∑_{1≤i<j≤n} y_{i}y_{j} =

= −4 [\frac{a_{2}}{a_{0}} + \frac{a_{1}}{na_{1}} (n − 1) ∑_{i=1}^{n} x_{i} + \frac{a_{1}^{2}}{n^{2}a_{0}^{2}} · \frac{n(n − 1)}{2}] = \frac{1}{a_{0}^{2}} Δ_{i}.

3 Austere Submanifolds

Remark 3.1. Obviously, not any minimal submanifold is austere. An austere submanifold with L_{i} = 0, ∀i = \overline{1, m}, for every base ξ_{1}, ..., ξ_{m} in T^{⊥}M is totally geodesic. For the submanifolds of a Riemannian manifold we have the following inclusions:

totally geodesic ⊂ austere ⊂ minimal.

In [3], with the notations established as above, is presented Theorem 3.1. If M is a proper slant submanifold of a Kaehlerian manifold (M̄, J, g), and if ∇F = 0, then M is austere. Remark 3.2. For an austere submanifold we have

(3.1)  L_{i}(p) = 2 max_{i=\overline{1,n}} ρ_{j}^{i}(p) = 2ρ_{max}^{i}(p),

and for every point p of an austere submanifold M ⊂ M̄ we have

(3.2)  \sqrt{( \begin{matrix}n \\ 2 \end{matrix} )^{−1} δ_{i}(p)} ≤ 2ρ_{max}^{i}(p) ≤ \sqrt{δ_{i}(p)}, ∀i = \overline{1, p},

(see Theorem 2.1). Theorem 3.2. Let M be a n-dimensional austere submanifold in the n+m dimensional Riemannian manifold (M̄, g). If for any orthonormal basis ξ_{1}, ..., ξ_{m} in T^{⊥}M we have

δ_{i}(p) = 0, ∀i = \overline{1, m}, ∀p ∈ M,

then M is totally geodesic. Proof. The relation (2.6) means L_{i}(p) = 0, ∀i = \overline{1, m} and ∀p ∈ M, and from (3.2) we have ρ_{max}^{i}(p) = 0, ∀p ∈ M, q.e.d. Following the theorem 3.1 we find Corollary 3.1. Let M be a n-dimensional proper slant submanifold of the n+mdimensional Kaehlerian manifold (M̄, J, g). If ∇F = 0 and for every basis ξ_{1}, ..., ξ_{m} in T^{⊥}M we have δ_{i} ≡ 0, ∀i = \overline{1, m}, then M is totally geodesic in M̄. Proof. The theorem 3.1 shows that M is an austere submanifold, and the theorem 3.2 yields that M is totally geodesic. Definition 3.1. Let ξ_{1}, ..., ξ_{m} be as above an orthonormal basis in T^{⊥}M, and the m shape discriminants, respectively δ_{1}, ...δ_{m}. We consider the mapping

δ : M → R^{m}, δ(p) = (δ_{1}(p), ..., δ_{m}(p)).

We call δ the principal discriminant of the submanifold M, corresponding to the basis ξ_{1}, ..., ξ_{m}.

Remark 3.3. If M is a totally geodesic submanifold, then δ = 0_{R^{p}}. The principal discriminant measures how far from the property of being totally geodesic is the submanifold M ⊂ M̄. Theorem 3.3. Let M be a submanifold of the Riemannian manifold M̄. If there exist i ∈ {1, ..., m} and p ∈ M such that δ_{i}(p) > 0, then M cannot be totally geodesic in M̄. Remark 3.4. The condition ||δ|| > 0 is an obstruction for the submanifold M to be totally geodesic.

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