FRENET FRAMES AND TODA SYSTEMS

A. V. RAZUMOV

Abstract. It is shown that the integrability conditions of the equations satisfied by the local Frenet frame associated with a holomorphic curve in a complex Grassmann manifold coincide with a special class of nonabelian Toda equations. A local moving frame of a holomorphic immersion of a Riemann surface into a complex Grassmann manifold is constructed and the corresponding connection coefficients are calculated.

1. Introduction

In the present paper we consider local differential geometry of a holomorphic curve ψ in the complex Grassmann manifold G^{k}(ℂ^{n}). It is well known that the simplest way to study such a curve is to consider a holomorphic local lift ξ of ψ to the space Mat(n, k; ℂ) of complex n×k matrices, defined on some open subset U of M. The natural object arising here is the so called Frenet frame being a mapping which associates a basis of ℂ^{n} with a point of U. We give a procedure of the construction of the Frenet frame and derive the equations which are satisfied by the corresponding ℂ^{n}-valued functions on M. Then we show that the obtained equations are closely connected to nonabelian Toda systems based on the Lie group GL(n, ℂ). The equations describing these systems are exactly integrable, so we get a possibility to construct holomorphic embeddings with prescribed properties. Note that our definition of the Frenet frame is different from the definition usually used for the problem under consideration, see, for example, [Li97]. Namely, we do not perform the complete orthogonalisation of the basis, and it is our definition that leads to an exactly integrable system. The paper is concluded with the explanation of how the Frenet frame can be used to construct a local moving frame associated with the mapping ψ and with the calculation of the corresponding connection coefficients.

There is a lot of papers discussing geometry of harmonic mappings of a Riemann surface into a projective space, see, for example, [EWo83, BWo92] and references therein. Since any holomorphic mapping from one Kähler manifold to another one is harmonic, most results of those paper are valid for holomorphic mappings. There are also some papers devoted to consideration of harmonic mappings from Riemann surfaces to Grassmann manifolds [EWo83, CWo85, BWo86, BSa87, CWo87, Wol88, Woo88, Val88, Uhl89]. In these papers the problems of classification and construction of harmonic mappings are mainly investigated. On the other hand, local differential geometry properties of holomorphic mappings from Riemann surfaces to Grassmann manifolds have not yet been discussed in a systematic way. In this respect we would like to mention the paper [Li97] which deals with differential geometry of holomorphic mappings from the two-sphere to a Grassmann manifold.

Note that the connection of abelian Toda systems based on the Lie group SL(n, ℂ) with the equations satisfied by the Frenet frame associated with an immersion of a Riemann surface into a projective space appeared to be useful for the study of W - geometries [GMa92, GMa93, Ger93]. Moreover, this connection allows to give a simple proof of the infinitesimal Plücker relations. The abelian Toda systems based on other semisimple Lie groups can be associated with equations satisfied by the Frenet frames of holomorphic curves in the corresponding flag manifolds. This leads to the so-called generalised infinitesimal Plücker relations [RSa94, GSa96]. The results of the present paper show that there are some relations between the geometric characteristics of the holomorphic curves associated with a holomorphic curve in a Grassmann manifold, but here we have no direct analogues of the Plücker relations. This problem requires an additional consideration. It is also very interesting to investigate the corresponding W -geometries.

2. Frenet frames

2.1. Holomorphic curves in Grassmann manifolds. We start with the description of the realisation of complex Grassmann manifolds which is convenient for our purposes. Let Mat(n, k; ℂ) denote the complex manifold of n × k complex matrices. Assume that n > k and denote by Mat(n, k; ℂ)^{×} the open submanifold of Mat(n, k; ℂ) formed by n × k matrices of rank k. The manifold Mat(n, k; ℂ)^{×} could be naturally identified with the set of all bases of k-dimensional subspaces of ℂ^{n}. Two points p, p′ ∈ Mat(n, k; ℂ)^{×} correspond to bases of the same subspace of ℂ^{n} if and only if there is an element g of the complex general linear group GL(k, ℂ) such that p′ = pg. Thus we have the free right action of the complex Lie group GL(k, ℂ) on Mat(n, k; ℂ)^{×} connecting different bases of the k-dimensional subspaces of ℂ^{n}. The orbit space here is the Grassmann manifold G^{k}(ℂ^{n}). The canonical projection π : Mat(n, k; ℂ)^{×} → G^{k}(ℂ^{n}) is holomorphic and Mat(n, k; ℂ)^{×} →^{π} G^{k}(ℂ^{n}) is a holomorphic principal GL(k, ℂ)-bundle.

Consider a holomorphic curve ψ : M → G^{k}(ℂ^{n}), where M is some connected Riemann surface. Any such a curve defines a holomorphic subbundle \underline{ψ} of rank k of the trivial holomorphic fibre bundle \underline{ℂ}^{n} = M × ℂ^{n}. Here the fiber over the point p ∈ M is ψ(p). The mapping ψ is said to be linearly full if the bundle \underline{ψ} is not contained in any proper trivial subbundle of \underline{ℂ}^{n}.

To describe geometry of the corresponding holomorphic curve in G^{k}(ℂ^{n}) it is convenient to use local holomorphic lifts of the mapping ψ to Mat(n, k; ℂ)^{×}. We say that a holomorphic mapping ξ : U → Mat(n, k; ℂ)^{×}, where U is an open subset of M, is a local holomorphic lift of ψ if it satisfies the relation

π ∘ ξ = ψ|_{U}.

Since our consideration is local, we will use local holomorphic lifts defined in coordinate neighbourhoods of M. The corresponding local coordinate will be denoted by z. Moreover, to make our notations consistent with those used in description of Toda systems, we denote z^{+} = z, z^{−} = z̄ and

∂_{−} = ∂/∂z, ∂_{+} = ∂/∂z̄.

Let ξ : U → Mat(n, k; ℂ)^{×} be a lift of a holomorphic mapping ψ : M → G^{k}(ℂ^{n}). Denote by f_{1}, …, f_{k} the holomorphic mappings from U to ℂ^{n} determined by the columns of the matrix-valued mapping ξ. For any p ∈ U the linear subspace W_{p} of ℂ^{n} spanned by the vectors f_{1}(p), …, f_{k}(p) and by the vectors given by the first and higher order derivatives of the functions f_{1}, …, f_{k} over z at the point p, is independent of the choice of the lift ξ and local coordinate z.

Proposition 2.1. For any points p and p′ of M one has

W_{p} = W_{p′}.

Proof. Let p be an arbitrary point of U and c = z(p). Expand the functions

f_{1} ∘ z^{−1}, …, f_{k} ∘ z^{−1} (2.1)

in the power series at c, which converges absolutely in some disc D(c, r) ⊂ z(U). Since the first and higher order derivatives of the functions f_{1} ∘ z^{−1}, …, f_{k} ∘ z^{−1} over z can be also expanded at c in the power series which absolutely converge in D(c, r_{c}), it is clear that for any p′ ∈ z^{−1}(D(c, r)) one has W_{p′} ⊂ W_{p}. From the other hand, if p′ ∈ z^{−1}(D(c, r/2)) then the functions (2.1) and their first and higher derivatives over z can be expanded at the point c′ = z(p′) in a power series which converge at least in the disc D(c′, r/2). Since p ∈ z^{−1}(D(c′, r/2)), one concludes that W_{p} ⊂ W_{p′}. Therefore, for all points p′ such that |z(p) − z(p′)| < r/2 one has W_{p′} = W_{p}.

Considering various local lifts of ψ we conclude that for any point p ∈ M there is an open neighbourhood U_{p} such that W_{p′} = W_{p} for all p′ ∈ U_{p}. Now the assertion of the Proposition follows from the connectedness of M.

Thus we have some constant subspace W of ℂ^{n} which characterises the curve ψ. The mapping ψ can be considered as a mapping from M to G^{k}(W), in this case it can be easily shown that ψ is linearly full. Therefore, without any loss of generality we can restrict ourselves to the consideration of linearly full mappings.

2.2. Some properties of vector-valued holomorphic functions. In this Section V is an n-dimensional complex vector space and U is an open connected subset of the complex plane ℂ. The results similar to ones discussed below in this Section for the case of polynomial mappings are proved in [Li95, Li97].

A mapping from U to V is called a V -valued function on U. Let f be a V -valued function on U. For any w ∈ V^{∗} one defines the complex function w(f) by

w(f)(c) = w(f(c)).

If g is a V^{∗}-valued function on U one defines the complex function g(f) by

g(f)(c) = g(c)(f(c)).

The function f is said to be holomorphic, if for any w ∈ V^{∗} the function w(f) is holomorphic.

Let {e_{i}} be a basis of V, {e^{i}} be the dual basis of V^{∗}, and f be a V -valued function on U. It is easy to verify that for any c ∈ U one has

f(c) = ∑_{i=1}^{n} e_{i} b^{i}(c),

where b^{i} = e^{i}(f). Here it is customary to write

f = ∑_{i=1}^{n} e_{i} b^{i}. (2.2)

It is clear that the function f is holomorphic if and only if the functions b^{i} are holomorphic.

If f is a holomorphic V -valued function on U and g is a holomorphic V^{∗}-valued function on U, then g(f) is a holomorphic function on U.

A point c ∈ U is called a zero of a holomorphic V -valued function f ≢ 0 on U, if f(c) = 0. A zero of a holomorphic V -valued function f ≢ 0 is a common zero of the functions b^{i} entering a representation of type (2.2). Here, in general, the order of the zero is different for different values of the index i, moreover, it depends on the choice of the basis {e_{i}}. Nevertheless, the minimal value of order does not depend on the choice of the basis {e_{i}} and it characterises the V -valued function f itself. The corresponding positive integer is called the order of zero of f.

A holomorphic V -valued function f ≢ 0 on U may have only a countable number of isolated zeros, and the set of zeros has no limit points in U.

If c ∈ U is a zero of order μ of a holomorphic V -valued function f ≢ 0 on U, one can represent f as

f = g(z − c)^{μ}.

Here g is a holomorphic V -valued function on U having the same zeros as f, except the zero at a.

Proposition 2.2. Let f ≢ 0 be a holomorphic V -valued function on U. There exists a holomorphic V -valued function g on U having no zeros in U and such that there is valid the representation

f = g d,

where d is a holomorphic function on U.

Proof. In accordance with the generalisation of the Weierstrass theorem [Mar77], there exists a holomorphic function d having the same zeros and with the same orders as f. It is easy to show that the function g = f/d satisfies the requirement of the Proposition.

The rank of a set {f_{1}, …, f_{k}} of holomorphic V -valued functions on U is defined as

rank{f_{1}, …, f_{k}} = max_{c∈U} rank{f_{1}(c), …, f_{k}(c)}.

The rank of a set {f_{1}, …, f_{k}} of holomorphic V -valued functions on U equals k if and only if f_{1} ∧ ⋯ ∧ f_{k} ≢ 0. We say that a set {f_{1}, …, f_{k}} of holomorphic V -valued functions on U is of constant rank if

rank{f_{1}(c), …, f_{k}(c)} = rank{f_{1}, …, f_{k}}

for any c ∈ U.

Proposition 2.3. Let the set {f_{1}, …, f_{k}} of holomorphic V -valued functions on U be of rank k. Then there exists a set {g_{1}, …, g_{k}} of V -valued holomorphic functions on U of constant rank k such that

f_{α} = ∑_{β=1}^{k} g_{β} d_{α}^{β}, α = 1, …, k,

where d_{α}^{β} are holomorphic functions on U.

Proof. We prove the Proposition by induction over k. The case k = 1 coincides with Proposition 2.2. Suppose that the Proposition is valid for some fixed k. Consider a set {f_{1}, …, f_{k}, f_{k+1}} of holomorphic V -valued functions on U having rank k + 1. It is clear that the rank of the set {f_{1}, …, f_{k}} equals k. Hence, one can find a set {g_{1}, …, g_{k}} of holomorphic V -valued functions on U, satisfying the requirement of the Proposition.

Taking into account Proposition 2.2, represent f_{k+1} in the form

f_{k+1} = g d,

where the holomorphic V -valued function g has no zeros, and consider the holomorphic ⋀^{k+1}(V)-valued function g_{1} ∧ ⋯ ∧ g_{k} ∧ g. If this function has no zeros, then we take g as g_{k+1} and get the set of functions {g_{1}, …, g_{k}, g_{k+1}} which satisfies the requirement of the Proposition.

Suppose now that the function g_{1} ∧ ⋯ ∧ g_{k} ∧ g has zeros at the points of the set {c_{ι}}_{ι∈I} and the zero at c_{ι} is of order μ_{ι}. Since g_{1}(c_{ι}) ∧ ⋯ ∧ g_{k}(c_{ι}) ∧ g(c_{ι}) = 0 and the vectors g_{1}(c_{ι}), …, g_{k}(c_{ι}) are linearly independent one has

g(c_{ι}) = ∑_{β=1}^{k} g_{β}(c_{ι}) b_{ι}^{β},

where b_{ι}^{β} are some complex numbers. Using the generalisation of the Weierstrass theorem and the generalisation of the Mittag–Leffler theorem [Mar77], one can show that there exist functions b^{β}, β = 1, …, k, which are holomorphic on U and have the property

b^{β}(c_{ι}) = b_{ι}^{β}.

Define the holomorphic V -valued function

h = g − ∑_{β=1}^{k} g_{β} b^{β}.

It is clear that this function has zeros only at the points of the set {c_{ι}}. Moreover, since

g_{1} ∧ ⋯ ∧ g_{k} ∧ η = g_{1} ∧ ⋯ ∧ g_{k} ∧ g,

the order of zero of h at the point c_{ι} is less or equal to the order of zero of g_{1}∧⋯∧g_{k}∧g. Represent h in the form

h = g′ d′,

where the holomorphic V -valued function g′ has no zeros. Then the equality

g_{1} ∧ ⋯ ∧ g_{k} ∧ g = g_{1} ∧ ⋯ ∧ g_{k} ∧ g′ d′

implies that for any ι ∈ I either the point c_{ι} is not a zero of the function g_{1}∧⋯∧g_{k}∧g′, or it is a zero of it but of order less than μ_{ι}. If the function g_{1} ∧ ⋯ ∧ g_{k} ∧ g′ has no zeros, we take g′ as g_{k+1} and from the equality

f_{k+1} = (∑_{β=1}^{k} g_{β} b^{β} + g′ d′) d

conclude that the set {g_{1}, …, g_{k}, g_{k+1}} satisfies the requirement of the Proposition. If it is again not the case, we repeat along the lines of this paragraph using the function g′ instead of g.

Since the order of a zero is always finite, after a finite number of steps we come to the set {g_{1}, …, g_{k}, g_{k+1}} satisfying the requirement of the Proposition.

From the proof of Proposition 2.3 it follows that the functions g_{1}, …, g_{k} can be chosen in such a way that the matrix D = (d_{α}^{β}) is upper triangular. Moreover, if the set {f_{1}, …, f_{l}}, l ≤ k, is of constant rank l, then we can choose g_{1} = f_{1}, …, g_{l} = f_{l}.

Proposition 2.4. Let {f_{1}, …, f_{k}}, k < n, be a set of holomorphic V -valued functions on U of constant rank k. Then there exists a set {f_{k+1}, …, f_{n}} of holomorphic V -valued functions on U such that the set {f_{1}, …, f_{k}, f_{k+1}, …, f_{n}} is of constant rank n on U.

Proof. Let c be some point of U. Denote e_{1} = f_{1}(c), …, e_{k} = f_{k}(c). The vectors e_{1}, …, e_{k} are linearly independent, and one can find vectors e_{k+1}, …, e_{n} such that {e_{1}, …, e_{k}} is a basis of V. Consider the set {f_{1}, …, f_{k}, e_{k+1}} of holomorphic V - valued functions on U. It is clear that the rank of this set equals k + 1. Applying arguments used in the proof of Proposition 2.3, one can show that there is a holomorphic V -valued function f_{k+1} on U such that the set {f_{1}, …, f_{k}, f_{k+1}} is of constant rank k + 1 on U. Using such a construction repeatedly we find a set {f_{1}, …, f_{n}} satisfying the requirement of the Proposition.

Proposition 2.5. Let {f_{1}, …, f_{n}} be a set of holomorphic V -valued functions on U of constant rank n. Then there exists a unique set {f^{1}, …, f^{n}} of holomorphic V^{∗}-valued functions on U such that

f^{i}(f_{j}) = δ_{j}^{i}, i, j = 1, …, n.

Proof. Let {e_{i}} be a basis of V, and {e^{i}} be the dual basis of V^{∗}. One has

f_{i} = ∑_{j=1}^{n} e_{j} b_{i}^{j},

where b_{i}^{j}, i, j = 1, …, n, are holomorphic functions on U. Since for any c ∈ U the vectors f_{1}(c), …, f_{n}(c) are linearly independent, the matrix (b_{i}^{j}(c)) is nondegenerate. Hence, there are holomorphic functions d_{j}^{i}, i, j = 1, …, n, on U such that for any c ∈ U one has

∑_{m=1}^{n} d_{m}^{i}(c) b_{j}^{m}(c) = δ_{j}^{i}, i, j = 1, …, n.

It is easy to get convinced that the holomorphic V^{∗}-valued functions

f^{i} = ∑_{j=1}^{n} d_{j}^{i} e^{j}, i = 1, …, n,

satisfy the requirement of the Proposition.

Proposition 2.6. Let {f_{1}, …, f_{k}} be a set of holomorphic V -valued functions on U of constant rank k, and g be a holomorphic V -valued function on U such that f_{1} ∧ ⋯ ∧ f_{k} ∧ g ≡ 0. Then there is a unique representation

g = ∑_{α=1}^{k} f_{α} b^{α},

where b^{α}, α = 1, …, k, are holomorphic functions on U.

Proof. For any c ∈ U the vectors f_{1}(c), …, f_{k}(c) are linearly independent, while the vectors f_{1}(c), …, f_{k}(c), g(c) are linearly dependent, therefore one has

g = ∑_{α=1}^{k} f_{α} b^{α},

where b^{α}, α = 1, …, k, are complex functions on U. Due to Proposition 2.4 one can construct holomorphic V -valued functions f_{k+1}, …, f_{n} such that the set {f_{1}, …, f_{n}} is of constant rank n. Let f^{i}, i = 1, …, n, be the holomorphic V^{∗}-valued functions satisfying the requirement of Proposition 2.5. Using these functions, one gets

b^{α} = f^{α}(g), α = 1, …, k.

Thus, the functions b^{α}, α = 1, …, k, are holomorphic.

Proposition 2.7. Let the set {f_{1}, …, f_{k}} of holomorphic V -valued functions on U be of rank l ≤ k. Then there is a set {g_{1}, …, g_{l}} of holomorphic V -valued functions on U of constant rank l such that there is valid the representation

f_{α} = ∑_{β=1}^{l} g_{β} d_{α}^{β}, α = 1, …, k, (2.3)

where d_{α}^{β}, α = 1, …, k, β = 1, …, l, are holomorphic functions on U.

If {g′_{1}, …, g′_{l}} is another set of holomorphic V -valued functions on U of constant rank l and for some holomorphic functions d′_{α}^{β} one has

f_{α} = ∑_{β=1}^{l} g′_{β} d′_{α}^{β}, α = 1, …, k,

then for some holomorphic functions c_{α}^{β}, α, β = 1, …, l, one has

g′_{α} = ∑_{β=1}^{l} g_{β} c_{α}^{β}, α = 1, …, l. (2.4)

Proof. Without loss of generality we can assume that the rank of the set {f_{1}, …, f_{l}} equals l. Thus, in accordance with Proposition 2.3 one can find a set {g_{1}, …, g_{l}} of holomorphic V -valued functions on U such that the set {g_{1}, …, g_{l}} is of constant rank l and (2.3) is valid for α = 1, …, l. It is not difficult to understand that for any α such that l < α ≤ k one has g_{1} ∧ ⋯ ∧ g_{l} ∧ f_{α} ≡ 0. Thus, due to Proposition 2.6 representation (2.3) is valid for all values of α.

To prove the second part of the Proposition, we suppose that the set {f_{1}, …, f_{l}} is again of rank l. Then one has

f_{1} ∧ ⋯ ∧ f_{l} = g_{1} ∧ ⋯ ∧ g_{l} det D̃, (2.5) f_{1} ∧ ⋯ ∧ f_{l} = g′_{1} ∧ ⋯ ∧ g′_{l} det D̃′, (2.6)

where D̃ and D̃′ are the holomorphic Mat(l, ℂ)-valued functions formed by the functions d_{α}^{β} and d′_{α}^{β}, α, β = 1, …, l, respectively. From (2.6) it follows that

f_{1} ∧ ⋯ ∧ f_{l} ∧ g′_{α} ≡ 0, α = 1, …, l.

Using this relation and taking into account (2.5), one concludes that

g_{1} ∧ ⋯ ∧ g_{l} ∧ g′_{α} ≡ 0, α = 1, …, l.

Now Proposition 2.6 implies that representation (2.4) is valid.

Proposition 2.8. Let {g_{1}, …, g_{k}} be a set of V -valued holomorphic functions on U of constant rank k, and {f_{1}, …, f_{l}} be a set of V -valued holomorphic functions on U such that the rank of the set {g_{1}, …, g_{k}, f_{1}, …, f_{l}} equals k + m, m ≤ l. Then there exist a set {g_{k+1}, …, g_{k+m}} of V -valued holomorphic functions on U such that the set {g_{1}, …, g_{k}, g_{k+1}, …, g_{k+m}} is of constant rank k + m and there is valid the representation

f_{α} = ∑_{β=1}^{k+m} g_{β}d_{α}^{β}, α = 1, …, l,

where d_{α}^{β}, α = 1, …, l, β = 1, …, k + m, are holomorphic functions on U.

Proof. The validity of the Proposition follows from the proof of Proposition 2.3, from the discussion given after that proof, and from the proof of Proposition 2.7.

2.3. Construction of Frenet frame. Let M be a Riemann surface, and U be an open subset of M. The rank of a holomorphic Mat(n, k; ℂ)-valued function ξ on U is defined as

rank ξ = max_{p∈U} rank ξ(p).

We say that a holomorphic Mat(n, k; ℂ)-valued function ξ on U is of constant rank if

rank ξ(p) = rank ξ

for each p ∈ U. A Mat(n, k; ℂ)-valued holomorphic function ξ on U generates the set {f_{1}, …, f_{k}} of ℂ^{n}-valued holomorphic functions on U determined by the columns of ξ. A Mat(n, k; ℂ)-valued holomorphic function ξ on U has rank l if and only if the corresponding set {f_{1}, …, f_{k}} is of rank l. A Mat(n, k; ℂ)-valued holomorphic function ξ on U is of constant rank if and only if the corresponding set {f_{1}, …, f_{k}} is of constant rank. Having these remarks in mind, we use the results of Section 2.2 for Mat(n, k; ℂ)-valued holomorphic functions.

Consider again a holomorphic curve ψ : M → G^{k}(ℂ^{n}) and some holomorphic lift ξ : U ⊂ M → Mat(n, k; ℂ)^{×} of ψ. The function ξ considered as a Mat(n, k; ℂ)-valued function is of constant rank k.

Since, in general, the function ξ will be the first member of some finite sequence of holomorphic matrix-valued functions, it is convenient to denote ξ_{0} = ξ and k_{0} = k. Consider the holomorphic Mat(n, 2k_{0}; ℂ)-valued function (ξ_{0} ∂ξ_{0}/∂z) on U. This function is of rank k_{0} + k_{1}, where 0 ≤ k_{1} ≤ k_{0}. In the case k_{1} = 0 due to Proposition 2.8 one has

∂_{−}ξ_{0} = ξ_{0} B_{00}, (2.7)

where B_{00} is a holomorphic Mat(k_{0}; ℂ)-valued function on U. Therefore, all the derivatives of ξ_{0} over z can be expressed via ξ_{0} by relation similar to (2.7). If k_{1} > 0, then again due to Proposition 2.8 there exists a holomorphic Mat(n, k_{1}; ℂ)-valued function ξ_{1} of constant rank k_{1} such that

∂_{−}ξ_{0} = ξ_{0} B_{00} + ξ_{1} B_{10}, (2.8)

where B_{00} is a holomorphic Mat(k_{0}; ℂ)-valued function and B_{10} is a holomorphic Mat(k_{1}, k_{0}; ℂ)-valued function. The linear subspaces of ℂ^{n} spanned by the columns of the matrix (ξ_{0}(p) ξ_{1}(p)), p ∈ U, do not depend on the choice of functions ξ_{0} and ξ_{1}. Actually, they are determined only by the mapping ψ. Using these subspaces, we define a holomorphic subbundle \underline{ψ}_{1} of \underline{ℂ}^{n} called the first osculating space of ψ, which generates a holomorphic curve ψ_{1} : M → G^{k_{0}+k_{1}}(ℂ^{n}) called the first associated curve of ψ.

Consider the holomorphic Mat(n, k_{0} + 2k_{1}; ℂ)-valued function (ξ_{0} ξ_{1} ∂ξ_{1}/∂z). It is of rank k_{0} + k_{1} + k_{2}, where 0 ≤ k_{2} ≤ k_{1}. Again, if k_{2} = 0, one concludes that the derivatives of ξ_{0} and ξ_{1} over z can be expressed via ξ_{0} and ξ_{1}. If k_{2} ≠ 0 one finds a holomorphic Mat(n, k_{2}; ℂ)-valued function ξ_{2} of constant rank k_{2} such that

∂_{−}ξ_{1} = ξ_{0} B_{01} + ξ_{1} B_{11} + ξ_{2} B_{21}.

After a finite number of steps we end up with the functions ξ_{0}, …, ξ_{t} satisfying the relations

∂_{−}ξ_{a} = ∑_{b=1}^{a+1} ξ_{b} B_{ba}, (2.9)

where B_{t+1,t} ≡ 0. Thus, all the derivatives of ξ_{0}, …, ξ_{t} over z can be expressed via ξ_{0}, …, ξ_{t}. Each step of the construction gives us the corresponding subbundle \underline{ψ}_{a}, a = 1, …, t, of \underline{ℂ}^{n} and we get the flag of the subbundles

\underline{ψ}_{0} ⊂ \underline{ψ}_{1} ⊂ ⋯ ⊂ \underline{ψ}_{t}.

Proposition 2.9. For any p ∈ M the fiber \underline{ψ}_{tp} coincide with the linear subspace V defined in Section 2.1.

Proof. The independence of \underline{ψ}_{tp} of p can be proved along the lines of the proof of Proposition 2.1. So we have just one subspace of ℂ^{n} which we denote by W′. It is quite clear that W ⊂ W′. Suppose that W is a proper subspace of W′. In this case there exists an element w ∈ ℂ^{n∗} such that w(v) = 0 for each v ∈ W, but w(u) ≠ 0 for some element of W′. Identifying ℂ^{n} with the space of complex n × 1 matrices and ℂ^{n∗} with the space of complex 1 × n matrices, we can write

w(v) = wv.

The condition w(v) = 0 for each v ∈ W is equivalent to the requirement

w ∂_{−}^{l}ξ = 0

for l = 0, 1, . . .. In particular, one has

w ξ = w ξ_{0} = 0.

Further, using (2.8) one obtains

w ∂_{−}ξ_{0} = w ξ_{1} B_{10} = 0.

Since the Mat(k_{1}, k_{0}; ℂ)-valued function B_{10} is of rank k_{1}, one has

w ξ_{1} = 0.

Similarly one shows that w ξ_{a} = 0 for all a = 1, …, t, but this contradicts to our supposition. Thus, W′ = W.

For a linearly full mapping ψ one has W = ℂ^{n}, therefore, \underline{ψ}_{t} = \underline{ℂ}^{n}. In any case the tth associated curve is trivial.

Starting from this point, we assume that the linear space ℂ^{n} is endowed with a positive definite hermitian scalar product

(v, u) = ∑_{i,j=1}^{n} \overline{v^{i}} h_{īj} u^{j},

that in matrix notation can be written as

(v, u) = v^{†}h u.

Here and below we denote by A^{†} the hermitian conjugate of the matrix A.

Now we proceed to the construction of what we call the Frenet frame associated with the lift ξ of the mapping ψ. Consider a Mat(n, ℂ)-valued function

Π_{0} = I_{n} − ξ_{0}(ξ_{0}^{†} h ξ_{0})^{−1}ξ_{0}^{†} h, (2.10)

where I_{n} is the n × n unit matrix. For any p ∈ U the matrix Π_{0}(p) is a matrix of the operator of the orthogonal projection to the orthogonal complement of \underline{ψ}_{0p} in ℂ^{n}, with respect to the canonical basis of ℂ^{n}. From (2.8) one immediately gets

Π_{0} ∂_{−}ξ_{0} = Π_{0} ξ_{1} B_{10}. (2.11)

Denoting

φ_{0} = ξ_{0}, φ_{1} = Π_{0} ξ_{1}, (2.12)

and

β_{0} = φ_{0}^{†} h φ_{0},

we can write (2.11) in the form

∂_{−}φ_{0} = φ_{0} β_{0}^{−1}∂_{−}β_{0} + φ_{1} B_{10}, (2.13)

where we used the relation

∂_{−}β_{0} = φ_{0}^{†} h ∂_{−}φ_{0}

which follows from the equality

∂_{+}φ_{0} = 0.

The subspaces spanned by the linear combinations of the columns of the matrix φ_{1}(p) do not depend on the choice of the lift ξ = ξ_{0} and the function ξ_{1}. These subspaces generate the subbundle \underline{φ}_{1} of \underline{ℂ}^{n}. Actually it is an orthogonal complement of the subbundle \underline{φ}_{0} = \underline{ψ}_{0} in \underline{ψ}_{1}. Thus, we have the following representation

\underline{ψ}_{1} = \underline{φ}_{0} ⊕ \underline{φ}_{1},

where the sum in the right hand side is orthogonal. Note that the orthogonality of the fibers of \underline{φ}_{0} and \underline{φ}_{1} is equivalent to the validity of the equality

φ_{0}^{†} h φ_{1} = 0.

Find now the equations satisfied by the function φ_{1}. Define the Mat(n, ℂ)-valued function Π_{1} by

Π_{1} = I_{n} − φ_{1}β_{1}^{−1}φ_{1}^{†}h,

where

β_{1} = φ_{1}^{†} h φ_{1}.

For any p ∈ U the matrix Π_{1}(p) is the matrix of the operator of the orthogonal projection to the orthogonal complement of \underline{φ}_{1p} in ℂ^{n}. Since the subspaces \underline{φ}_{0p} and \underline{φ}_{1p} are orthogonal, we have

Π_{0} Π_{1} = Π_{1} Π_{0}. (2.14)

12 A. V. RAZUMOV

Using relations (2.12), (2.14) and the equality

∂_{−}ξ_{1} = ξ_{0} B_{01} + ξ_{1} B_{11} + ξ_{2} B_{21},

one gets Π_{2} ∂_{−}φ_{1} = Π_{1} ∂_{−}Π_{0} ξ_{1} + Π_{1} Π_{0} ξ_{2} B_{21}. (2.15) It follows from (2.10) and (2.13) that

∂_{−}Π_{0} = −φ_{1} B_{10} β_{0}^{−1}φ_{0}^{†} h.

Therefore, denoting

φ_{2} = Π_{1} Π_{0} ξ_{2},

we rewrite (2.15) as ∂_{−}φ_{1} = φ_{1} β_{1}^{−1}φ_{1}^{†} h ∂_{−}φ_{1} + φ_{2} B_{21}. (2.16) Similarly one gets

∂_{+}Π_{0} = h^{−1} (∂_{−}Π_{0})^{†} h = φ_{0} β_{0}^{−1}D_{01}φ_{1}^{†} h,

where

D_{01} = −B_{10}^{†}

is an antiholomorphic Mat(k_{1}, k_{2}; ℂ)-valued function. Hence,

∂_{+}φ_{1} = φ_{0} β_{0}^{−1}D_{01} β_{1}.

This equation, in particular, implies that

∂_{−}β_{1} = φ_{1}^{†} h ∂_{−}φ_{1}.

Therefore, equation (2.16) can be written as

∂_{−}φ_{1} = φ_{1} β_{1}^{−1}∂_{−}β_{1} + φ_{2} B_{21}.

As before, the columns of the matrices φ_{2}(p) generate a subbundle \underline{φ}_{2} and we have the orthogonal decomposition

\underline{ψ}_{2} = \underline{φ}_{0} ⊕ \underline{φ}_{1} ⊕ \underline{φ}_{2}.

In general, defining inductively the functions

β_{a} = φ_{a}^{†} h φ_{a},

the projectors

Π_{a} = I_{n} − φ_{a}β_{a}^{−1}φ_{a}^{†} h

and the functions

φ_{a+1} = Π_{a} ⋯ Π_{0} ξ_{a+1},

we get the orthogonal decompositions

\underline{ψ}_{a} = ⨁_{b=0}^{a} \underline{φ}_{b}.

Note that the sequence of subbundles \underline{φ}_{1}, …, \underline{φ}_{t} and the sequence of the mappings from M to the corresponding Grassmann manifolds are partial cases of the so called harmonic sequences of subbundles and mappings intensively used to classify and construct harmonic mappings from a Riemann surface to a Grassmann manifold.

Theorem 2.1. The functions φ_{a}, a = 0, …, t, satisfy the equations

∂_{−}φ_{a} = φ_{a} β_{a}^{−1}∂_{−}β_{a} + φ_{a+1} B_{a+1,a}, (2.17) ∂_{+}φ_{a} = φ_{a−1} β_{a−1}^{−1}D_{a−1,a} β_{a}, (2.18)

where B_{t+1,t} ≡ 0, D_{−1,0} ≡ 0 and

D_{a−1,a} = −(B_{a,a−1})^{†}. (2.19)

Proof. Assume that that we proved the validity of equations (2.17), (2.18) for all a ≤ b < t. Using these equations for a = b, one has

∂_{−}Π_{b} = −φ_{b+1} B_{b+1,b} β_{b}^{−1}φ_{b}^{†}h + φ_{b} B_{b,b−1} β_{b−1}^{−1} φ_{b−1}^{†}h,
∂_{+}Π_{b} = −φ_{b−1} β_{b−1}^{−1} D_{b−1,b} φ_{b}^{†}h + φ_{b} β_{b}^{−1}D_{b,b+1} φ_{b+1}^{†}h.

Now differentiating the definition of φ_{b+1} over z^{−} and z^{+}, we can easily show that equations (2.17), (2.18) are valid for a = b + 1.

Consider the Mat(n, ℂ)-valued mapping φ = (φ_{0} ⋯ φ_{t}). For any p ∈ U the matrix φ(p) is nondegenerate. Denote by f_{1}, …, f_{n} the ℂ^{n}-valued functions determined by the columns of φ. For any p ∈ U the vectors f_{1}(p), …, f_{n}(p) are linearly independent and form a basis in ℂ^{n}. We call the set of mappings f_{1}, …, f_{n}, or the mapping φ which generates this set, the Frenet frame of ψ associated with the lift ξ.

3. Connection with Toda systems

3.1. ℤ-graded Lie algebras. To get the equations describing a Toda system [LSa92, RSa94, RSa97a, RSa97b] one starts with a complex Lie group G whose Lie algebra 𝔤 is endowed with a ℤ-gradation

𝔤 = ⨁_{m∈ℤ}𝔤_{m}.

Introduce the following subalgebras of 𝔤

𝔥̃ = 𝔤_{0},  𝔫̃_{−} = ⨁_{m<0} 𝔤_{m},  𝔫̃_{+} = ⨁_{m>0} 𝔤_{m}.

and denote by H̃ and Ñ_{±} the connected Lie subgroups of G corresponding to the subalgebras 𝔥̃ and 𝔫̃_{±} respectively.

Suppose that H̃ and Ñ_{±} are closed subgroups of G and, moreover,

H̃ ∩ Ñ_{±} = {e},  Ñ_{−} ∩ Ñ_{+} = {e},
Ñ_{−} ∩ H̃Ñ_{+} = {e},  Ñ_{−}H̃ ∩ Ñ_{+} = {e},

where e is the unit element of G. This is, in particular, true for the finite-dimensional complex reductive Lie groups, see, for example, [Hum75]. Let the set Ñ_{−}H̃Ñ_{+} be dense in G, then for any element g which belongs to Ñ_{−}H̃Ñ_{+} one can write the following unique decomposition

g = n_{−}hn_{+}^{−1}, (3.1)

where n_{−} ∈ Ñ_{−}, h ∈ H̃ and n_{+} ∈ Ñ_{+}. This is again true for the finite-dimensional complex reductive Lie groups. Decomposition (3.1) is called the Gauss decomposition.

There is a simple classification of possible ℤ-gradations for complex semisimple Lie algebras, see, for example, [GOV94, RSa97b]. In this case for any ℤ-gradation of a such an algebra 𝔤, there exists a unique element q ∈ 𝔤 which has the following property. An element x ∈ 𝔤 belongs to the subspace 𝔤_{m} if and only if [q, x] = mx. The element q is called the grading operator.

Let 𝔤 be a semisimple Lie algebra of rank r. Denote by h_{i} and i = 1, …, r, some set of the Cartan generators of 𝔤. For any set of r nonnegative numbers l_{i} the element

q = ∑_{i,j=1}^{r} h_{i}(k^{−1})_{ij}l_{j}, (3.2)

where k = (k_{ij}) is the Cartan matrix of 𝔤, is the grading operator of some ℤ-gradation of 𝔤. The numbers l_{i} can be considered as the labels assigned to the vertices of the corresponding Dynkin diagram. The well-known canonical gradation of 𝔤 arises when one chooses all the numbers l_{i} equal to 2. Actually, if two sets of labels are connected by an automorphism of the Dynkin diagram, we get two ℤ-gradations connected by an ‘external’ automorphism of 𝔤. If it is not the case, then different sets of labels give ℤ-gradations which cannot be connected by an automorphism of 𝔤.

If all the labels of the Dynkin diagram of a semisimple Lie algebra 𝔤 are different from zero, then the subgroup 𝔤_{0} coincides with the Cartan subalgebra 𝔥. In this case the subgroup H̃ is abelian and we obtain the so called abelian Toda equations. In all other cases the subgroup H̃ is nonabelian and the corresponding Toda equations are called nonabelian.

If we deal with a reductive Lie algebra, we choose as the grading operator any grading operator of its maximal semisimple subalgebra.

3.2. Toda equations and their general solution. Let M be a simply connected complex one dimensional manifold. Consider a reductive complex Lie group G whose Lie algebra 𝔤 is endowed with a ℤ-gradation. Let l be a positive integer, such that the grading subspaces 𝔤_{m} for −l < m < 0 and 0 < m < l are trivial, and c_{−} and c_{+} be some fixed mappings from M to the subspaces 𝔤_{−l} and 𝔤_{+l}, respectively, such that c_{−} is holomorphic and c_{+} is antiholomorphic. Restrict ourselves to the case when G is a matrix Lie group. In this case the Toda equations are the matrix partial differential equations of the form

∂_{+}(γ^{−1}∂_{−}γ) = [c_{−}, γ^{−1}c_{+}γ], (3.3)

where γ is a mapping from M to H̃. These equations are the zero curvature condition for the connection ω = ω_{−}dz^{−}+ω_{+}dz^{+} on the trivial principal G-bundle M ×G, where

ω_{−} = γ^{−1}∂_{−}γ + c_{−}, ω_{+} = γ^{−1}c_{+}γ.

Since M is simply connected, then there exists a mapping φ : M → G such that

∂_{−}φ = φ(γ^{−1}∂_{−}γ + c_{−}), ∂_{+}φ = φγ^{−1}c_{+}γ.

To obtain the general solution of Toda equations one can use the following procedure [LSa92, RSa94, RSa97a, RSa97b]. Choose some mappings γ_{±} from M to H̃ such that γ_{−} is holomorphic and γ_{+} is antiholomorphic. Then integrate the equations

μ_{±}^{−1}∂_{±}μ_{±} = γ_{±}c_{±}γ_{±}^{−1}, ∂_{∓}μ_{±} = 0. (3.4)

The Gauss decomposition (3.1) induces the corresponding decomposition of mappings from M to G. In particular, one obtains

μ_{+}^{−1}μ_{−} = ν_{−}ην_{+}^{−1}, (3.5)

where the mapping η takes values in H̃, and the mappings ν_{±} take values in Ñ_{±}. It can be shown that the mapping

γ = γ_{+}^{−1}ηγ_{−} (3.6)

satisfies the Toda equations, and any solution to these equations can be obtained by the described procedure. Note that for the corresponding mapping φ one has the following expression [RSa94, RSa97a, RSa97b]

φ = gμ_{+}ν_{−}ηγ_{−} = gμ_{−}ν_{+}γ_{−}, (3.7)

where g is an arbitrary constant element of G.

We will be interested in the so called hermitian solutions of Toda equations [RSa94, RSa97b]^{1}. Let the Lie algebra 𝔤 be endowed with some antilinear antiautomorphism σ which can be extended to the antiholomorphic antiautomorphism Σ of the Lie group G. Suppose that

σ(𝔤_{m}) = 𝔤_{−m}, m ∈ ℤ.

In this case one has

Σ(H̃) = H̃, Σ(Ñ_{±}) = Ñ_{∓}.

Assume that the mappings c_{±} entering the Toda equations are subjected to the condition

σ(c_{−}) = −c_{+}.

In this case, if γ is a solution to Toda equations, then Σ ∘ γ is also a solution of the same equations, and we can consider solutions of Toda equations having the property

Σ ∘ γ = γ. (3.8)

To get such solutions using the general procedure of integration of the Toda equations described above, we should start with the mappings γ_{±} which satisfy the relation

γ_{+} = Σ ∘ γ_{−}^{−1}

and choose the solutions of (3.4) for which

μ_{+} = Σ ∘ μ_{−}^{−1}.

1Actually in [RSa94, RSa97b] we use the term ‘real solutions’.

It can be shown that in such a way we get all the solutions of the Toda equations satisfying (3.8). We call such solutions hermitian. The mapping φ corresponding to a hermitian solution of Toda equations satisfies the relation

(Σ ∘ φ)Σ(g)gφ = γ,

where g is the element of G entering (3.7).

3.3. Nonabelian Toda equations associated with Lie group GL(n, ℂ) and Frenet frames. In this Section, generalising the consideration of [RSa97c], we give the general form of nonabelian Toda equations based on the Lie group GL(n, ℂ) and establish their connection to the equations of the Frenet frame.

The Lie algebra 𝔤𝔩(n, ℂ) of the Lie group GL(n, ℂ) is reductive and can be represented as the direct product of the simple Lie algebra 𝔰𝔩(n, ℂ) of rank n − 1 and a one dimensional Lie algebra isomorphic to 𝔤𝔩(1, ℂ) and composed by the n × n complex matrices which are multiplies of the unit matrix. Consider the general ℤ-gradation of 𝔰𝔩(n, ℂ) arising when we choose the labels of the corresponding Dynkin diagram as follows

0  0 s_{1} 0  0 s_{2} 0  0 s_{t} 0  0 ^{k_{0}−1}  ^{k_{1}−1}  ^{k_{t}−1}

It is convenient to take as a Cartan subalgebra for 𝔰𝔩(n, ℂ) the subalgebra consisting of diagonal n × n matrices with zero trace. Here the standard choice of the Cartan generators is

(h_{i})_{kl} = δ_{ki}δ_{li} − δ_{k,i+1}δ_{l,i+1}, i = 1, …, n − 1.

With such a choice of Cartan generators, using (3.2), we obtain the following block matrix form of the grading operator

q = ( \begin{matrix}ρ_{0}I_{k_{0}} & 0 & ⋯ & 0 & 0 \\ 0 & ρ_{1}I_{k_{1}} & ⋯ & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & ⋯ & ρ_{t−1}I_{k_{t−1}} & 0 \\ 0 & 0 & ⋯ & 0 & ρ_{t}I_{k_{t}} \end{matrix} ) ,

where

ρ_{a} = \frac{1}{n} (− ∑_{b=1}^{a} s_{b} ∑_{c=0}^{b−1} k_{c} + ∑_{b=a+1}^{t} s_{b} ∑_{c=b}^{t} k_{c}) .

We use this grading operator to define a ℤ-gradation of the Lie algebra 𝔤𝔩(n, ℂ). It is not difficult to describe the arising grading subspaces of 𝔤𝔩(n, ℂ) and the relevant subgroups of GL(n, ℂ). To this end we consider an element x of 𝔤𝔩(n, ℂ) as a t × t block matrix (x_{ab})_{a,b=0}^{t}, where x_{ab} is k_{a}×k_{b} matrix. For fixed a ≠ b, the block matrices x having only the block x_{ab} different from zero belong to the grading subspace 𝔤_{m} with

m = ∑_{c=a+1}^{b} s_{c}, a < b, m = − ∑_{c=b+1}^{a} s_{c}, a > b.

The block-diagonal matrices form the grading subspace 𝔤_{0}.

Using the same block matrix representation for the elements of GL(n, ℂ), we see that the subgroup H̃ consists of all block-diagonal nondegenerate matrices, and the subgroups Ñ_{−} and Ñ_{+} consist, respectively, of all block lower and upper triangular matrices with unit matrices on the diagonal.

The equations related to Frenet frame arise in the case when all integers s_{a}, a = 1, …, t, are equal to 2. In this case the mappings c_{±} have to take values in the subspaces 𝔤_{±2} and their general form is

c_{−} = ( \begin{matrix}0 & 0 & ⋯ & 0 & 0 \\ B_{10} & 0 & ⋯ & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 0 & 0 \\ 0 & 0 & ⋯ & B_{t,t−1} & 0 \end{matrix} ) , c_{+} = ( \begin{matrix}0 & D_{01} & ⋯ & 0 & 0 \\ 0 & 0 & ⋯ & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 0 & D_{t−1,t} \\ 0 & 0 & ⋯ & 0 & 0 \end{matrix} )

Parametrise the mapping γ as

γ = ( \begin{matrix}β_{0} & 0 & ⋯ & 0 & 0 \\ 0 & β_{1} & ⋯ & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & ⋯ & β_{t−1} & 0 \\ 0 & 0 & ⋯ & 0 & β_{t} \end{matrix} ) .

The corresponding Toda equations are

∂_{+}(β_{a}^{−1}∂_{−}β_{a}) = −β_{a}^{−1}D_{a,a+1}β_{a+1}B_{a+1,a} + B_{a,a−1}β_{a−1}D_{a−1,a}β_{a}, (3.9)

where B_{t+1,t} ≡ 0 and D_{−1,0} ≡ 0.

Represent the mapping φ entering (3.8) in the block form φ = (φ_{0} ⋯ φ_{t}), where φ_{a} is a Mat(n, k_{a}; ℂ)-valued function. Then it is clear that relations (3.8) literally coincide with equations (2.17), (2.18). Thus, the integrability conditions of the equations satisfied by the Frenet frame are the Toda equations (3.9).

Not any solution of equations (3.9) gives us a mapping φ describing the Frenet frame. First of all we should consider the Toda systems for which relations (2.19) are satisfied. These relations are equivalent to the equality (c_{−})^{†} = −c_{+}. Moreover, for the Frenet frame one has β_{a}^{†} = β_{a}, which is equivalent to γ^{†} = γ. Hence, we should consider only hermitian solutions to the Toda equations corresponding to the antiholomorphic antiautomorphism of GL(n, ℂ) generated by the ordinary hermitian conjugation of matrices. Finally, for the Frenet frame one has φ^{†} h φ = γ. Therefore, constructing the mapping φ with the help of (3.7) one should take the constant element g of G satisfying the relation g^{†}g = h.

4. Geometrical interpretation of Frenet frames

4.1. Invariant metrics on principal fibre bundles. Let P →^{π} N be a principal fibre G-bundle, and g^{∗} be a metric on P which is invariant with respect to the standard right action of G on P. The metric g^{∗} determines a connection on P for which the horizontal subspace H_{p} ⊂ T_{p}(P), p ∈ P, is defined as the orthogonal complement to the vertical subspace V_{p} ⊂ T_{p}(P) tangent to the fiber through p. For any vector field X on N there is a unique horizontal vector field X^{∗} on P satisfying the relation

dπ ∘ X^{∗} = X ∘ π. (4.1)

The vector field X^{∗} is called the horizontal lift of X. The vector field X^{∗} is invariant with respect to the standard right action of G on P. Below we call a vector field on P invariant with respect to the standard right action of G simply an invariant vector field. Any invariant horizontal vector field on P is the horizontal lift of a unique vector field on N. By definition, for any vector field X on N and a function f on N one has

π^{∗}(X(f)) = X^{∗}(π^{∗}f). (4.2)

Denote by X^{⊥} the component of the vector field X on P orthogonal to the fibers, that is actually the horizontal component of X. For any vector fields X and Y on N the relation

[X, Y]^{∗} = [X^{∗}, Y^{∗}]^{⊥} (4.3)

is valid.

The metric g^{∗} determines a metric g on N defined by the equality

π^{∗}(g(X, Y)) = g^{∗}(X^{∗}, Y^{∗}). (4.4)

Proposition 4.1. The Levi–Civita connections ∇^{∗} and ∇ associated with g^{∗} and g satisfy the equality

(∇_{Y} X)^{∗} = (∇_{Y^{∗}}^{∗}X^{∗})^{⊥}.

Proof. The validity of the assertion of the Proposition follows immediately from the Koszul formula

2g(∇_{Y} X, Z) = Y (g(X, Z)) + X(g(Z, Y)) − Z(g(Y, X))
− g(Y, [X, Z]) + g(X, [Z, Y]) + g(Z, [Y, X])

after taking into account definition (4.4) and relations (4.2), (4.3).

Let ψ be an immersion of a manifold M to N. A mapping X : N → T (M) is said to be a vector field on ψ if

pr_{M} ∘X = ψ,

where pr_{M} is the canonical projection of T (M) to M. A vector field X′ defined on some open subset of M is called an extension of a vector field X on ψ if

X′ ∘ ψ = X.

Here and below writing relations containing extensions of vector fields we imply the appropriate restrictions of the domains.

The covariant derivative ∇_{Y} X of the vector field X on ψ along the vector field Y on M is defined as

∇_{Y} X = ∇_{(dψ∘Y)′}X′ ∘ ψ, (4.5)

where X′ and (dψ ∘ Y)′ are some arbitrary extensions of the vector fields X and dψ ∘ Y.

Let now ξ : U ⊂ M → P be a local lift of ψ. The mapping ξ is an immersion of U to P. We call a vector field X on ξ vertical if for any p ∈ U the vector X_{p} is tangent to the fiber through p. Similarly, a vector field X on ξ is said to be horizontal if for any p ∈ U the tangent vector X_{p} is horizontal, or, in other words, orthogonal to the fibre through ξ(p).

Denote the induced connection generated by ψ also by ∇.

Proposition 4.2. For any vector field X on ψ and a vector field Y on M the relation

∇_{Y} X = dπ ∘ ∇_{(dψ∘Y)^{′∗}}^{∗}X^{′∗} ∘ ξ

is valid.

Proof. From Proposition 4.1 it follows that

(∇_{(dψ∘Y)′}X′)^{∗} = (∇_{(dψ∘Y)^{′∗}}^{∗}X^{′∗})^{⊥}.

Relation (4.1) gives

∇_{(dψ∘Y)′}X′ ∘ π = dπ ∘ ∇_{(dψ∘Y)^{′∗}}^{∗}X^{′∗}.

The statement of the Proposition follows now from definition (4.5).

Proposition 4.3. Let X be a horizontal vector field on ξ. There is an extension X′ of X such that X′ is horizontal and invariant.

Proof. The vector field dπ ∘ X is a vector field on ψ. Let (dπ ∘ X)′ be an arbitrary extension of dπ ∘ X. Consider the vector field

X′ = (dπ ∘ X)^{′∗}. (4.6)

This vector field is, by definition, horizontal and invariant. Let us show that it is an extension of X. Indeed, from (4.6) one gets

dπ ∘ X′ ∘ ξ = (dπ ∘ X)′ ∘ ψ = dπ ∘ X.

Since, X′ ∘ ξ and X are horizontal vector fields, then X′ ∘ ξ = X. Thus, X′ is an extension of X which satisfies the requirements of the Proposition.

Below by an extension of a vector field X on ξ we mean an extension of X which satisfies the requirements of Proposition 4.3.

Proposition 4.4. Let X be a horizontal vector field on ξ, and Y be a vector field on M. For the vector field Z = dπ ∘ X on ψ one has

∇_{Y} Z = dπ ∘ ∇_{(dξ∘Y)^{⊥′}}^{∗}X′ ∘ ξ.

Proof. Since X′ is horizontal and invariant, there is a unique vector field Z′ on N such that

Z^{′∗} = X′.

It can be easily shown that Z′ is an extension of Z. From Proposition 4.2 it follows that

∇_{Y} Z = dπ ∘ ∇_{(dψ∘Y)^{′∗}}^{∗}X′ ∘ ξ.

Further, the vector field (dξ ∘ Y)^{⊥′} is horizontal and invariant. Therefore, there is a unique vector field Y′ such that

(dξ ∘ Y)^{⊥′} = Y^{′∗}.

It is not difficult to demonstrate that Y′ is an extension of the vector field dψ ∘ Y. Thus, we see that the assertion of the Proposition is true.

4.2. Construction of a local moving frame. Let us return to the case of holomorphic curves in Grassmann manifolds. Denote by w_{α}^{i}, i = 1, …, n, α = 1, …, k, the standard coordinate functions on Mat(n, k; ℂ) and their restrictions to Mat(n, k; ℂ)^{×}. A generic vector field on Mat(n, k; ℂ)^{×} has a unique representation

X = ∑_{i=1}^{n} ∑_{α=1}^{k} (X_{α}^{i} \frac{∂}{∂w_{α}^{i}} + Xī ; ᾱ \frac{∂}{∂w̄ī ; ᾱ} ) .

Recall that Mat(n, k; ℂ)^{×} is the principal GL(k, ℂ)-bundle over G^{k}(ℂ^{n}). The vertical subspaces are generated by the Killing vector fields corresponding to the right action of the Lie group GL(k, ℂ) on Mat(n, k; ℂ)^{×}. The vector fields

K_{α}^{β} = ∑_{i=1}^{n} w_{α}^{i} \frac{∂}{∂w_{β}^{i}} , α, β = 1, …, k,

and

K̄β̄ ; ᾱ = ∑_{i=1}^{n} w̄ī ; ᾱ \frac{∂}{∂w̄ī ; β̄}, α, β = 1, …, k,

form bases in the spaces of the holomorphic and antiholomorphic Killing vector fields. A vector field X on Mat(n, k; ℂ)^{×} is invariant with respect to the right action of GL(k, ℂ) if and only if

[K_{α}^{β}, X] = 0,  [\overline{K}β̄ ; ᾱ, X] = 0,  α, β = 1, …, k,

that is equivalent to

∑_{i=1}^{n} w_{α}^{i} \frac{∂X_{γ}^{j}}{∂w_{β}^{i}} = X_{α}^{j}δ_{γ}^{β},  ∑_{i=1}^{n} w_{α}^{i} \frac{∂Xj̄ ; γ̄}{∂w_{β}^{i}} = 0,

∑_{i=1}^{n} w̄ī ; ᾱ \frac{∂X_{γ}^{j}}{∂w̄ī ; β̄} = 0,  ∑_{i=1}^{n} w̄ī ; ᾱ \frac{∂Xj̄ ; γ̄}{∂w̄ī ; β̄} = Xj̄ ; ᾱδβ̄ ; γ̄.

To construct an invariant metric on Mat(n, k; ℂ)^{×} define first the Mat(k, ℂ)-valued function Δ = (Δ_{ᾱβ}) on Mat(n, k; ℂ)^{×} by

Δ_{ᾱβ} = ∑_{i,j=1}^{n} \overline{w}ī ; ᾱh_{īj}w_{β}^{j}.

It is quite clear that the metric g^{∗} on Mat(n, k; ℂ)^{×} given by

g^{∗} = ∑_{i,j=1}^{n} ∑_{α,β=1}^{k} d\overline{w}ī ; ᾱ ⊗ h_{īj} dw_{β}^{j} (Δ^{−1})^{βᾱ} + ∑_{i,j=1}^{n} ∑_{α,β=1}^{k} h_{j̄i} dw_{α}^{i} (Δ^{−1})^{αβ̄} ⊗ d\overline{w}ī ; β̄

is invariant with respect to the right action of GL(k, ℂ) on Mat(n, k; ℂ)^{×}. Hence, we can define the metric g on G^{k}(ℂ^{n}) generated by g^{∗}. This metric is proportional to the standard Kähler metric on G^{k}(ℂ^{n}).

Proposition 4.5. For any horizontal vector fields X and Y on Mat(n, k; ℂ)^{×} one has

(∇_{Y} X)^{⊥} = ∑_{i,j=1}^{n} ∑_{α,β=1}^{k} Y_{β}^{j} (\frac{∂X_{α}^{i}}{∂w_{β}^{j}} \frac{∂}{∂w_{α}^{i}} + \frac{∂Xī ; ᾱ}{∂w_{β}^{j}} \frac{∂}{∂w̄ī ; ᾱ} )^{⊥}

+ ∑_{i,j=1}^{n} ∑_{α,β=1}^{k} Y j̄ ; β̄(\frac{∂X_{α}^{i}}{∂w̄j̄ ; β̄} \frac{∂}{∂w_{α}^{i}} + \frac{∂Xī ; ᾱ}{∂w̄j̄ ; β̄} \frac{∂}{∂w̄ī ; ᾱ} )^{⊥} .

Proof. A vector field X on Mat(n, k; ℂ)^{×} is orthogonal to the fibers if and only if

∑_{i,j=1}^{n} \overline{w}j̄ ; β̄h_{j̄i}X_{α}^{i} = 0,  ∑_{i,j=1}^{n} Xī ; ᾱh_{īj}w_{β}^{j} = 0.

The Christoffel symbols of the metric g^{∗} are

Γ_{αjk}^{iβγ} = −\frac{1}{2} ∑_{m=1}^{n} ∑_{δ=1}^{k} (δ_{j}^{i}δ_{α}^{γ}(Δ^{−1})^{βδ̄}\overline{w}m̄ ; δ̄h_{m̄k} + δ_{k}^{i}δ_{α}^{β}(Δ^{−1})^{γδ̄}\overline{w}m̄ ; δ̄h_{m̄j}) ,

Γ_{αj̄k}^{iβ̄γ} = Γ_{αkj̄}^{iγβ̄} = −\frac{1}{2} (∑_{m=1}^{n} δ_{k}^{i}(Δ^{−1})^{γβ̄}h_{j̄m}w_{α}^{m} − ∑_{δ=1}^{k} δ_{α}^{γ}h_{j̄k}w_{δ}^{i}(Δ^{−1})^{δβ̄}) ,

Γ\begin{matrix}ī & β̄ & γ̄ \\ ᾱ & j̄ & k̄ \end{matrix} = \overline{Γ_{αjk}^{iβγ}},  Γ_{ᾱjk̄}^{īβγ̄} = Γ_{ᾱk̄j}^{īγ̄β} = \overline{Γ_{αj̄k}^{iβ̄γ}},  Γ_{αj̄k̄}^{iβ̄γ̄} = \overline{Γ_{ᾱjk}^{īβγ}} = 0.

Using the above relations one sees that the assertion of the Proposition is true.

For an arbitrary vector field X on ξ one has the representation

X = ∑_{i=1}^{n} ∑_{α=1}^{k} [X_{α}^{i} (\frac{∂}{∂w_{α}^{i}} ∘ ξ) + Xī ; ᾱ (\frac{∂}{∂w̄ī ; ᾱ} ∘ ξ)] ,

where X_{α}^{i} and Xī ; ᾱ are some functions on M.

Let φ be the Frenet frame of ψ associated with the lift ξ. Define the set of vector fields Ẽ_{i}^{α}, i = 1, …, n, α = 1, …, k, on ξ by

Ẽ_{i}^{α} = ∑_{j=1}^{n} φ_{i}^{j} (\frac{∂}{∂w_{α}^{j}} ∘ ξ) .

The vector fields Ẽ_{β}^{α}, α, β = 1, …, k, are vertical, while the vector fields Ẽ_{μ}^{α}, α = 1, …, k, μ = k + 1, … n, are horizontal.

The vector fields

E_{μ}^{α} = dπ ∘ Ẽ_{μ}^{α}, α = 1, …, k, μ = k + 1, …, n,

form a local basis in the space of vector fields on ψ. Therefore, one has

∇_{∂_{±}}E_{μ}^{α} = ∑_{ν=k+1}^{n} ∑_{β=1}^{k} E_{ν}^{β}(Λ_{±})_{βμ}^{να}.

Rewrite the equations (2.17) and (2.18) as

∂_{±}f_{i} = ∑_{j=1}^{n} f_{j}(λ_{±})_{i}^{j}

where the ℂ^{n}-valued functions f_{1}, …, f_{n} are generated by the columns of the mapping φ.

Theorem 4.1. The connection coefficients (Λ_{±})_{βμ}^{να} are connected with the functions (λ_{±})^{νμ} by the relations

(Λ_{−})_{βμ}^{να} = δ_{β}^{α}(λ_{−})_{μ}^{ν} − (β_{0}^{−1}∂_{−}β_{0})_{β}^{α}δ_{μ}^{ν}, (Λ_{+})_{βμ}^{να} = δ_{β}^{α}(λ_{+})_{μ}^{ν}.

Proof. The statement of the Theorem follows from Propositions 4.4 and 4.5.

In conclusion of this Section we give the expressions for the metrics induced on M by the mappings ψ_{a}, a = 0, …, t − 1. From the equations satisfied by the Frenet frame it follows immediately that for the metric g_{a} induced on ath associated curve one has

g_{a}(∂_{+}, ∂_{−}) = − tr(β_{a}^{−1}D_{a,a+1}β_{a+1}B_{a+1,a}) = tr(β_{a}^{−1}B_{a+1,a}^{†}β_{a+1}B_{a+1,a}).

Toda equations (3.9) give

∂_{+}∂_{−} ln det β_{a} = g_{a}(∂_{+}, ∂_{−}) − g_{a−1}(∂_{+}, ∂_{−}).

This equality implies that the function ln(det β_{0} ⋯ det β_{a}) is a Kähler potential of the metric g_{a}.

Acknowledgments

It is a pleasure to thank Yu. I. Manin and M. V. Saveliev for their interest to the work and for fruitful discussions. The author is grateful to the Max–Planck–Institut für Mathematik in Bonn where this work was completed for the warm hospitality and financial support. This work was also partially supported by the Russian Foundation for Basic Research under grant no. 95–01–00125, and by the INTAS grant no. 96–690.

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Institute for High Energy Physics, 142284 Protvino, Moscow Region, Russia.

  E-mail:  razumov@mx.ihep.su