Proceedings Ist International Meeting on Geometry and Topology Braga (Portugal)

Public. Centro de Matemática da Universidade do Minho p. 39–42, 1998

On k-minimal submanifolds

S. Carter

This is an account of joint work with U.Dursun. Details can be found in [1, 2, 3].

Let M^{m} be a smooth, connected, m-dimensional manifold without boundary, and let f : M^{m} → R^{p} be a smooth immersion. I will consider immersions for which the focal set has certain symmetries. For example, in the case m = 2, in each normal plane the focal set is symmetrical about its base point if and only if the immersion is minimal. If m > 2, minimality can be interpreted as saying that, for all x ∈ M, on each normal line through f(x) the focal points with base x balance out (strictly speaking it is the principal curvatures that balance out). Letting N_{x}(f) and T_{x}(f) denote the normal plane and tangent plane to f at x, the technical condition is that ∀x ∈ M, ∀n ∈ N_{x}(f), trace A_{n} = 0, where A_{n} : T_{x}(f) → T_{x}(f) is the shape operator of f in direction n. It is this interpretation of minimality that I will use.

Now let Λ(f) be a line subbundle of N(f), the normal bundle of f. I will give a condition which will imply that, for each x ∈ M, the focal points in N_{x}(f) balance out along lines in N_{x}(f) perpendicular to Λ_{x}(f), the fibre of Λ(f). So, for m = 2, the focal set in N_{x}(f) has reflexive symmetry about Λ_{x}(f). This certainly needs trace A_{n} = 0 for all n perpendicular to Λ_{x}(f). Further, assuming that Λ(f) is parallel and u is a local parallel section, then f_{u} : M → R^{m+d} defined by f_{u}(x) = f(x) + u(x) is such that, for each x ∈ M, the normal planes and focal sets of f and f_{u} at x coincide. Then the required condition is that the trace of the shape operator of f_{u} in direction n is zero for all normals n perpendicular to Λ_{x}(f). This can be expressed in terms of the shape operator of f, to give trace(1 − A_{u})^{−1}A_{n} = 0. If this condition holds for all u for which f_{u} is an immersion (so (1 − A_{u}) is non-singular) I will say that f is 1-minimal. This can be generalized for symmetry about planes.

Definition 1 Let Λ(f) be a parallel, flat k-plane subbundle of N(f). The immersion f is k-minimal with respect to Λ(f) if ∀x ∈ M, ∀n ∈ N_{x}(f) with n ⊥ Λ_{x}(f), ∀u ∈ Λ_{x}(f) with 1 − A_{u} non-singular, trace(1 − A_{u})^{−1}A_{n} = 0.

40 S. Carter

Here is an equivalent definition.

Definition 2 The immersion f is k-minimal with respect to Λ(f) if ∀x ∈ M, ∀n ∈ N_{x}(f) with n ⊥ Λ_{x}(f), ∀u ∈ Λ_{x}(f), ∀ℓ = 0, 1, …, (m − 1), … trace(A_{u})^{l}A_{n} = 0.

Yet another equivalent statement can be obtained by considering Ω = {(x, u) ∈ Λ(f) : 1 − A_{u} is non-singular} and the immersion F : Ω → R^{m+d} defined by F (x, u) = f(x) + u.

Theorem 1([2]) The immersion f is k-minimal with respect to Λ(f) if and only if F is minimal.

The image of F is “locally ruled” in the sense that it is locally foliated by k-planes. The conditions on Λ(f) ensure a close relationship between the focal sets of f and F. It would be interesting to consider k-plane subbundles Λ(f) of N(f) which, locally, are minimal (in the Euclidean space) without any other conditions on Λ(f), but then the focal sets of f and F may not be so closely related. For example, a curve in R^{3} with nowhere zero curvature satisfies the condition with Λ_{x}(f) spanned by the principal normal at f(x). But the surface obtained by taking lines through the principal normals will not, in general, be minimal. If the curve is a helix then the surface obtained is a helicoid which is minimal.

Examples More details are given in [3].

(1) If f : M → R^{p} is pseudo-umbilical and H/ ∥ H ∥ is parallel (where H is the mean curvature vector) then f is 1-minimal with respect to the line bundle determined by H.

(2) Let f : S^{1} × S^{1} × S^{1} → R^{6} be defined by

f(α, β, γ) = (a cos α, a sin α, b cos β, b sin β, c cos γ, c sin γ)

where a > 0, b > 0, c > 0. Then f is 1-minimal if a = b = c, 2-minimal if two of a, b, c are equal and otherwise is 3-minimal.

Now if f is k-minimal with respect to Λ(f) then, from the equivalent definition, trace A_{n} = 0, ∀n ⊥ Λ_{x}(f). It follows that, ∀x ∈ M, the mean curvature vector H(x) ∈ Λ_{x}(f). In the 1-minimal case if the mean curvature vector H is not zero at x, then Λ_{x}(f) is the line determined by this vector, H/ ∥ H ∥ is parallel and, ∀n ⊥ H, trace(A_{H})^{l}A_{n} = 0, ℓ = 1, …, m − 1. Immersions for which trace A_{H}A_{n} = 0 ∀n ∈ N_{x}(f) with ⟨H, n⟩ = 0 have been studied by a number of people.

Definition 3 An immersion f : M^{m} → R^{p} is a Chen immersion (A-immersion) if trace A_{H}A_{n} = 0, ∀x ∈ M and ∀n ∈ N_{x}(f) with ⟨H, n⟩ = 0.

On k-minimal submanifolds 41

These immersions were introduced by B.Y.Chen in [4] and have been developed by, among others, L.Gheysens, P.Verheyen and L.Verstraelen [5], C.S.Houh [6], S.J.Li [7], and B.Rouxel [8]. It is immediate from the above comment that a 1-minimal immersion is Chen. For an immersion f : M^{2} → R^{p} with H/ ∥ H ∥ parallel then f is 1-minimal if and only if f is Chen. All known examples of Chen immersions of higher dimensional manifolds with H/ ∥ H ∥ parallel are 1-minimal but it seems very unlikely that these ideas are the same.

Finally I will describe a method for generating many examples of k-minimal immersions. Start with a pseudoumbilical immersion f : M^{m} → R^{p} (or, more generally, let f be 1-minimal) with u = H/ ∥ H ∥ parallel. Take any immersion g : W^{w} → R^{1+w}, g(y) = (g_{0}(y), …, g_{w}(y)), with unit normal field ψ : W → R^{1+w} and include R^{p} in R^{p+w} so that f can be thought of as an immersion into R^{p} × R^{w}. Then, at each x ∈ M, put the image of g into the 1 + w-subplane of the normal plane to f at x spanned by H(x) and R^{w}. More formally, define h : M × W → R^{p+w} by

h(x, y) = (f(x) + g_{0}(y)u(x), g_{1}(y), …, g_{w}(y)).

If M and W are compact, by replacing g by ϵg if necessary, for sufficiently small ϵ, h is an immersion. Then μ : M × W → R^{p+w} defined by

μ(x, y) = (ψ_{0}(y)u(x), ψ_{1}(y), …, ψ_{w}(y))

is a parallel normal field for h. It can be shown that h is 1-minimal with respect to the line bundle determined by μ [1]. This construction can be repeated, replacing f by h and, possibly, using a different g. A similar construction can be done if g has codimension k, which gives h is k-minimal. Even more generally we can start with f being k-minimal. These results are summarized in the following theorem.

Theorem 2([2]) Let f : M^{m} → R^{p} be an immersion with an orthonormal set of parallel normal fields u_{1}, …, u_{k} (1 ≤ k ≤ p − m), let q ≥ k and let g : W^{w} → R^{q}, g(y) = (g_{1}(y), …, g_{q}(y)), be an immersion with an orthonormal set of parallel normal fields ψ_{1}, … ψ_{q−w} such that ∀(x, y) ∈ M × W, 1 − ∑_{i=1}^{k} g_{i}(y)A_{u_{i}} is nonsingular, where A_{u_{i}} : T_{x}(f) → T_{x}(f) is the shape operator of f. Let h : M × W → R^{p+q−k} be defined by

h(x, y) = (f(x) + ∑_{i=1}^{k} g_{i}(y)u_{i}(x), g_{k+1}(y), …, g_{q}(y)).

Then h is an immersion and for j = 1, …, q − w, μ_{j} : M × W → R^{p+q−k} defined by

μ_{j}(x, y) = (∑_{i=1}^{k}(ψ_{j})_{i}(y)u_{i}(x), (ψ_{j})_{k+1}(y), …, (ψ_{j})_{q}(y))

is a parallel normal field for h and μ_{1}, …, μ_{q−w} is orthonormal. Let Λ(f) be the k-plane subbundle of N(f) with fibre

Λ_{x}(f) = span{u_{1}(x), …, u_{k}(x)}

and let Λ̃(h) be the (q − w)-plane subbundle of N(h) with fibre

Λ̃_{(x,y)}(h) = span{μ_{1}(x, y), …, μ_{q−w}(x, y)}.

Then, if f is k-minimal with respect to Λ(f), h is (q − w)-minimal with respect to Λ̃(h).

References

[1] S.Carter and U.Dursun, Partial tubes and Chen submanifolds, to appear in J. Geometry.

[2] S.Carter and U.Dursun, On generalised Chen and k-minimal immersions, Beiträge zur Algebra und Geometrie 38 (1997) 125-134.

[3] S.Carter and U.Dursun, Isoparametric and Chen submanifolds. Geometry and Topology of Submanifolds VIII, ed. F.Dillen, B.Komrakov, U.Simon, I.Van de Woestyne and L.Verstraelen, World Scientific Press, Singapore, (1996) 41-45.

[4] B.Y.Chen, Pseudo-umbilical submanifolds of a Riemannian manifold of constant curvature II, J.Math.Soc.Japan, 25 (1) (1973), 105-114.

[5] L.Gheysens, P.Verheyen and L.Verstraelen, Characterization and Examples of Chen Submanifolds, J.Geometry, 20 (1983), 47-62.

[6] C.S.Houh, On spherical A-submanifolds, Chinese J.Math. 2 (1974) 128-134.

[7] S.J.Li, Null 2-type Chen surfaces, Glasgow Math.J. 37 (1995) 233-242.

[8] B.Rouxel, A-submanifolds in Euclidean space, Kodai Math. J. 4 (1981), 181-188.

Sheila Carter

School of Mathematics University of Leeds Leeds, LS2 9JT United Kingdom