MATHEMATICS

P. I. SHVEIKIN

INVARIANT CONSTRUCTIONS ON AN \(m\)-DIMENSIONAL SURFACE IN AN \(n\)-DIMENSIONAL AFFINE SPACE

(Presented by Academician I. G. Petrovskii on 14 IV 1958)

With the current point \(\Lambda\) of the surface there is associated an arbitrary frame \(\vec{\Lambda}, E_\alpha\) \((\alpha,\beta=1,2,\ldots,n)\). The surface is given by the equation \(d\vec{\Lambda}=\omega^a\Lambda_a^\alpha E_\alpha\) \((a,b,c,d,e,f=1,2,\ldots,m)\), where \(\omega^\alpha\) are Pfaff forms determining the group of analytic transformations of the parametrization. Prolongations of this equation lead to a sequence of fields of fundamental objects of the surface. For every \(w\) the fundamental object \(\{\Lambda\}_w\) of order \(w\) consists of the components \(\Lambda_a^\alpha, \Lambda_{a^1a^2}^{\alpha_1},\ldots,\Lambda_{a^1,\ldots,a^w}^{\alpha}\), symmetric in the lower indices. It is known \((^1)\) that by a field of a fundamental object of sufficiently high order one may cover a field with any generating object. The aim of the paper is the construction and study of such coverings. The work is carried out by the invariant-group method of G. F. Laptev \((^1)\).

In the constructions the following geometric objects with constant components are used:
\(\mathcal{B}_{a^1\ldots a^r}^{b^1\ldots b^r}\equiv\delta_{(a^1}^{b^1}\cdots\delta_{a^r)}^{b^r}\),
\(\mathcal{B}_{a^1\ldots a^r}^{b^1\ldots b^s}\equiv0,\ r\geqslant1,\ s\ne r\);
\(\mathcal{T}_{a^1\ldots a^r,b}^{b^1\ldots b^s,c^1\ldots c^{r+1-s}}\equiv \mathcal{B}_{(a\ldots a^s}^{b^1\ldots b^s}\mathcal{B}_{a^{s+1}\ldots a^r)b}^{c^1\ldots c^{r+1-s}}\);
\(\mathcal{E}_{\alpha_1\alpha_2\ldots\alpha_n}\) is the unit \(n\)-vector;
\(\mathcal{E}_{a^1a^2\ldots a^m}\) is the unit \(m\)-vector;
\(\mathcal{E}_{a_1^1\ldots a_1^r,\ a_2^1\ldots a_2^r,\ldots,\ a_{m_r}^1\ldots a_{m_r}^r}\) is a relative tensor, symmetric in the indices belonging to one series and skew-symmetric with respect to the series, where
\[ m_r\equiv\frac{(m+r-1)!}{r!(m-1)!}. \]

1. Relative tensors are constructed
   \[ H_{\alpha_1\ldots\alpha_q} = \Lambda_1^{\beta_1^1\ldots\beta_{m_1}^1} \cdots \Lambda_{p-1}^{\beta_1^{p-1}\ldots\beta_{m_{p-1}}^{p-1}} \mathcal{E}_{\beta_1^1\ldots\beta_{m_1}^1\ldots\beta_1^{p-1}\ldots\beta_{m_{p-1}}^{p-1}\alpha_1\ldots\alpha_q}, \]
   \[ K_{a_1^1\ldots a_1^p,\ldots,a_q^1\ldots a_q^p} = \Lambda_{a_1^1\ldots a_1^p}^{\alpha_1} \cdots \Lambda_{a_q^1\ldots a_q^p}^{\alpha_q} H_{\alpha_1\ldots\alpha_q}, \]
   where
   \[ \Lambda_r^{\beta_1\ldots\beta_{m_r}} \equiv \Lambda_{a_1^1\ldots a_1^r}^{\beta_1} \cdots \Lambda_{a_{m_r}^1\ldots a_{m_r}^r}^{\beta_{m_r}} \mathcal{E}^{a_1^1\ldots a_1^r,\ldots,a_{m_r}^1\ldots a_{m_r}^r}, \]
   \[ q=n-(m_1+m_2+\cdots+m_{p-1}) \]
   and \(p\) is the number such that
   \[ m_1+\cdots+m_{p-1}<n\leq m_1+\cdots+m_p . \]
   The vanishing of the first (second) of these tensors is equivalent to a decrease in the dimension of the osculating plane of order \(p-1\) \((p)\).

2. A nonzero relative invariant cannot be covered by the object \(\{\Lambda\}_{p-1}\), and therefore the problem of covering it by the object \(\{\Lambda_p\}\) is of interest. In the case when \(q\) is either equal to \(2\), or is a divisor of \(m\),

is nonzero, the relative invariant

\[ K=L_{a_{11}^{1}\ldots a_{11}^{p+p},\,\ldots,\,a_{q1}^{1}\ldots a_{q1}^{p+p}}\cdots L_{a_{1m}^{1}\ldots a_{1m}^{p+p},\,\ldots,\,a_{qm}^{1}\ldots a_{qm}^{p+p}} \times \mathcal E^{a_{111}^{1},\ldots,a_{1i\ell}^{1}}\cdots \mathcal E^{a_{q1}^{p+p},\ldots,a_{qm}^{p+p}}, \]

where

\[ L_{a_{1}^{1}\ldots a_{1}^{p+p},\,\ldots,\,a_{q}^{1}\ldots a_{q}^{p+p}} = \mathfrak B_{a_{1}^{1}\ldots a_{1}^{p+p}}^{b_{1}^{1}\ldots b_{1}^{p}c_{1}^{1}\ldots c_{1}^{p}} \cdots \mathfrak B_{a_{q}^{1}\ldots a_{q}^{p+p}}^{b_{q}^{1}\ldots b_{q}^{p}c_{q}^{1}\ldots c_{q}^{p}} K_{b_{1}^{1}\ldots b_{1}^{p},\,\ldots,\,b_{q}^{1}\ldots b_{q}^{p}}^{c_{1}^{1}\ldots c_{1}^{p},\,\ldots,\,c_{q}^{1}\ldots c_{q}^{p}} . \]

In the case when \(x(x=m_p-q)\) is a divisor of \(m\), the relative invariant

\[ K^{*}=L_{a_{11}^{1}\ldots a_{11}^{p+p},\,\ldots,\,a_{x1}^{1}\ldots a_{x1}^{p+p}} \cdots L_{a_{1m}^{1}\ldots a_{1m}^{p+p},\,\ldots,\,a_{xm}^{1}\ldots a_{xm}^{p+p}} \times \mathcal E_{a_{11}^{1}\ldots a_{1m}^{1}}\cdots \mathcal E_{a_{x1}^{p+p}\ldots a_{xm}^{p+p}}, \]

where

\[ L_{1^{1}\ldots 1^{p+p},\,\ldots,\,x^{1}\ldots x^{p+p}}^{a^{1}\ldots a^{p+p},\,\ldots,\,a^{1}\ldots a^{p+p}} = \mathfrak B_{b_{1}^{1}\ldots b_{1}^{p}c_{1}^{1}\ldots c_{1}^{p}}^{a_{1}^{1}\ldots a_{1}^{p+p}} \cdots \mathfrak B_{b_{x}^{1}\ldots b_{x}^{p}c_{x}^{1}\ldots c_{x}^{p}}^{a_{x}^{1}\ldots a_{x}^{p+p}} \times K^{b_{1}^{1}\ldots b_{1}^{p},\,\ldots,\,b_{x}^{1}\ldots b_{x}^{p}} K^{c_{1}^{1}\ldots c_{1}^{p},\,\ldots,\,c_{x}^{1}\ldots c_{x}^{p}}, \]

\[ K^{b_{1}^{1}\ldots b_{1}^{p},\,\ldots,\,b_{x}^{1}\ldots b_{x}^{p}} = \mathcal E^{b_{1}^{1}\ldots b_{1}^{p},\,\ldots,\,b_{x}^{1}\ldots b_{x}^{p},\,c_{1}^{1}\ldots c_{1}^{p},\,\ldots,\,c_{q}^{1}\ldots c_{q}^{p}} K_{c_{1}^{1}\ldots c_{1}^{p},\,\ldots,\,c_{q}^{1}\ldots c_{q}^{p}} . \]

In the case when \(q=m_p\), the relative invariant

\[ K^{**}=K_{a_{1}^{1}\ldots a_{1}^{p},\,\ldots,\,a_{q}^{1}\ldots a_{q}^{p}} \mathcal E^{a_{1}^{1}\ldots a_{1}^{p},\,\ldots,\,a_{q}^{1}\ldots a_{q}^{p}} . \]

The relative invariant \(K\) (similarly \(K^{*}, K^{**}\)) can be represented in the following forms:

\[ K=\frac{1}{m_{1}}\Lambda_{\alpha}^{a}V_{a}^{\alpha} =\frac{1}{m_{2}}\Lambda_{\alpha}^{a_{1}a_{2}}V_{a}^{a_{1}a_{2}} =\cdots =\frac{1}{m_{p-1}}\Lambda_{a_{1}\ldots a_{p-1}}^{\alpha}V_{\alpha}^{a_{1}\ldots a_{p-1}} = \frac{1}{q}\Lambda_{a_{1}\ldots a_{p}}^{\alpha}V_{\alpha}^{a_{1}\ldots a_{p}} . \]

Here \(V_{\alpha}^{a_{1}\ldots a_{r}}\) are certain polynomials in the components of the object \(\{\Lambda\}_{p}\). Their aggregate forms a geometric object. A sequence of geometric objects is also formed by the quantities

\[ M_{a_{1}\ldots a_{r}}^{b_{1}\ldots b_{s}} = \frac{1}{K}\Lambda_{a_{1}\ldots a_{r}}^{\alpha}V_{\alpha}^{b_{1}\ldots b_{s}} . \]

For example, \(M_{a_{1}\ldots a_{p}}^{b_{1}\ldots b_{p}}\) is a tensor.

1. The basis of the subsequent constructions is a sequence of normal objects, which correspond to the objects of connections of higher degrees of V. Hlavatý \((^{2})\). The normal object \(\{n\}_{\omega}\) consists of components
   \(n_{a_{1}a_{2}}^{a}\), \(n_{a_{1}a_{2}a_{3}}^{a}\), \(\ldots\), \(n_{a_{1}\ldots a\omega}^{a}\), symmetric in the lower indices. For \(\{n\}_{2}\) the formula has been obtained

\[ n_{a_{1}a_{2}}^{a} = -\,W_{a_{1}a_{2},\,b}^{a,\,b_{1}b_{2}} M_{b_{1}b_{2}c_{1}\ldots c^{p-1}}^{bc_{1}\ldots c^{p-1}}, \]

in which \(W_{a_{1}a_{2},\,b}^{a,\,b_{1}b_{2}}\) are quantities uniquely determined by the linear system

\[ \mathfrak T_{d^{1}d^{2}e^{1}\ldots e^{p-1},\,a}^{a_{1}a_{2},\,c_{1}\ldots c^{p}} M_{c_{1}\ldots c^{p}}^{d^{1}e^{1}\ldots e^{p-1}} W_{a^{1}a^{2},\,b}^{a,\,b^{1}b^{2}} = \mathfrak B_{d^{1}d^{2}}^{b^{1}b^{2}}\mathfrak B_{b}^{d}. \]

(The nondegeneracy of this system has been verified in the case when \(q\) is either equal to \(m_p\), or is a divisor of \(m\).) To cover the following normal objects, a group of formulas recurrent with respect to the index \(r\) (for arbitrary \(s\)) is given:

\[ \mathfrak{K}^{a,b^1\ldots b^s}_{a^1\ldots a^s,b} = -\mathfrak{B}^{b^1\ldots b^s}_{a^1\ldots a^s}\mathfrak{B}^a_b, \qquad \mathfrak{K}^{a,b^1\ldots b^s}_{a^1\ldots a^{s+r},b} = \mathfrak{B}^{b^1\ldots b^s}_{a^1\ldots a^{s+r},b} \,\mathfrak{a}^{e^1\ldots e^{r+1}}_{e^1\ldots e^{r+1}}, \]

\[ m^{b^1\ldots b^s}_{a^1\ldots a^s} = \mathfrak{B}^{b^1\ldots b^s}_{a^1\ldots a^s}, \]

\[ m^{b^1\ldots b^s}_{a^1\ldots a^{s+r}} = -\sum_{u=2}^{r+1}\frac{u-1}{r}\, n^c_{c^1\ldots c^u} \sum_{v=u}^{r+1} m^{e^1\ldots e^{s+v-1}}_{a^1\ldots a^{s+r}}\, \mathfrak{F}^{d^1\ldots d^v,b^1\ldots b^s}_{e^1\ldots e^{s+v-1},d}\, \mathfrak{K}^{*\,d,c^1\ldots c^u}_{d^1\ldots d^v,d}, \]

\[ n^a_{a^1\ldots a^{r+2}} = -\mathfrak{D}^{a,b^1\ldots b^{r+2}}_{(p-r-1)\,a^1\ldots a^{r+2},b} \sum_{s=p-r-1}^{p} \mathop{m}^{*\,bc^1\ldots c^{p-r-1}}_{d^1\ldots d^s} \left( M^{d^1\ldots d^s}_{b^1\ldots b^{r+2}c^1\ldots c^{p-r-1}} + M^{d^1\ldots d^s}_{e^1\ldots e^p} m^{e^1\ldots e^p}_{b^1\ldots b^{r+2}c^1\ldots c^{p-r-1}} \right) + \]

\[ \quad +\mathfrak{D}^{a,b^1\ldots b^{r+2}}_{(p-r-1)\,a^1\ldots a^{r+2},b} \sum_{v=2}^{r+1}\frac{v-1}{r+1} n^f_{f^1\ldots f^v} \sum_{u=v}^{r+2} \mathfrak{F}^{e^1\ldots e^u,d^1\ldots d^{p+2-u}}_{b^1\ldots b^{r+2}c^1\ldots c^{p-r-1},e} \times \]

\[ \quad \times \mathop{m}^{*\,bc^1\ldots c^{p-r-1}}_{d^1\ldots d^{p+2-u}} \mathop{\mathfrak{K}}^{*\,e,f^1\ldots f^v}_{e^1\ldots e^u,f} + \]

\[ \quad +\mathfrak{D}^{a,b^1\ldots b^{r+2}}_{(p-r-1)\,a^1\ldots a^{r+2},b} \mathop{m}^{*\,bc^1\ldots c^{p-r-1}}_{d^1\ldots d^p} m^{d^1\ldots d^p}_{b^1\ldots b^{r+2}c^1\ldots c^{p-r-1}}, \qquad 1\le r\le p-2, \]

\[ n^a_{a^1\ldots a^{r+2}} = -\sum_{v=1}^{p} M^{b^1\ldots b^v}_{a^1\ldots a^{r+2}} \mathop{m}^{*\,a}_{b^1\ldots b^v}, \qquad r\ge p-1. \]

Here \(\mathop{\mathfrak{K}}^{*}\), \(\mathop{m}^{*}\), and \(\mathfrak{D}_{(s)}\) are the unique solutions of the nondegenerate linear systems

\[ \sum_{v=s}^{t} \mathfrak{K}^{a,e^1\ldots e^v}_{a^1\ldots a^t,e} \mathop{\mathfrak{K}}^{*\,e,b^1\ldots b^s}_{e^1\ldots e^v,b} = \mathfrak{B}^{b^1\ldots b^s}_{a^1\ldots a^t}\mathfrak{B}^{a}_{b}, \]

\[ \sum_{v=s}^{t} m^{e^1\ldots e^v}_{a^1\ldots a^t} \mathop{m}^{*\,b^1\ldots b^s}_{e^1\ldots e^v} = \mathfrak{B}^{b^1\ldots b^s}_{a^1\ldots a^t}, \]

\[ \mathfrak{F}^{c^1\ldots c^s,e^1\ldots e^r}_{d^1\ldots d^s\,e^1\ldots e^r,c}\, \mathfrak{D}^{c,b^1\ldots b^s}_{(r)\,c^1\ldots c^s,b} = \mathfrak{B}^{b^1\ldots b^s}_{a^1\ldots a^s}\mathfrak{B}^{a}_{b}. \]

1. The following lemma has been proved, which makes it possible to construct envelopes of some geometric objects by normal objects:

If the system

\[ dX^{b^1\ldots b^s}_{a^1\ldots a^r} = X^{b^1\ldots b^s}_{ca^2\ldots a^r}\omega^c_{a^1} +\cdots+ X^{b^1\ldots b^s}_{a^1\ldots a^{r-1}c}\omega^c_{a^r} - X^{cb^2\ldots b^s}_{a^1\ldots a^r}\omega^{b^1}_{c} -\cdots- \]

\[ \cdots - X^{b^1\ldots b^{s-1}c}_{a^1\ldots a^r}\omega^{b^s}_{c} + \sum_{t=2}^{w} X^{b^1\ldots b^s,c^1\ldots c^t}_{a^1\ldots a^r,c}\, \omega^c_{c^1\ldots c^t} + X^{b^1\ldots b^s}_{a^1\ldots a^r;c}\omega^c, \qquad \omega^c=0, \]

in which \(\omega^a_{a^1\ldots a^v}\) are invariant forms of the group of analytic transformations of the parametrization, \(s\ne r\), and the coefficients \(X^{b^1\ldots b^s,c^1\ldots c^t}_{a^1\ldots a^r,c}\) are known functions of the components of the normal object \(\{n\}_w\), is completely integrable, then the set of functions

\[ X^{b^1\ldots b^s}_{a^1\ldots a^r} = \frac{1}{r-s} \sum_{t=2}^{w}(t-1)n^c_{c^1\ldots c^t} \sum_{v=t}^{w} X^{b^1\ldots b^s,e^1\ldots e^v}_{a^1\ldots a^r,e}\, \mathop{\mathfrak{K}}^{*\,e,c^1\ldots c^t}_{e^1\ldots e^v,c} \]

is its unique solution depending only on the components of the normal objects.

1. An invariant furnishing of the surface is given in the form of the linear space
   \(\{e_{a_1a_2},\ldots,e_{a_1\ldots a_{p-1}}, M^{c_1\ldots c_p}_{a_1\ldots a_p}e_{c_1\ldots c_p}\}\), where
   \(e_{a_1\ldots a_r}=T^\alpha_{a_1\ldots a_r}E_\alpha\),

\[ T^\alpha_{a_1\ldots a_r} = \sum_{s=}^{r} m^{c_1\ldots c_s}_{a_1\ldots a_r}\Lambda^\alpha_{c_1\ldots c_s}. \]

It is shown that every tensor embraced by the fundamental object of the surface is embraced by the tensors \(T^\alpha_{a_1\ldots a_r}\).

1. The following property (see (3)) of the tensors is indicated:

\[ T^{b_1\ldots b_s}_{a_1\ldots a_{p+1}} \equiv \frac{1}{K}T^\alpha_{a_1\ldots a_{p+1}} \sum_{r=s}^{p} {}^{*}m^{b_1\ldots b_s}_{c_1\ldots c_r}V^{c_1\ldots c_r}_{\alpha}, \qquad s=1,2,\ldots,p, \]

\[ Q^a_{a_1\ldots a_r,b} = \sum_{t=1}^{r-1} {}^{*}m^a_{e_1\ldots e_t} m^{e_1\ldots e_t}_{a_1\ldots a_r;b} + \sum_{t=2}^{r+1} {}^{*}m^a_{e_1\ldots e_{t-1}b} m^{e_1\ldots e_{t-1}}_{a_1\ldots c^1}, \qquad r=2,3,\ldots,p: \]

the surface on which these tensors are equal to zero, and only such a surface, has the form

\[ \vec{\Lambda} = c+\sum_{r=1}^{p}\frac{1}{r!}c_{a_1\ldots a_r}s^{a_1}\cdots s^{a_r}. \]

Here \(c, c_{a_1\ldots a_r}\) are constant vectors, and the vectors \(c_{a_1\ldots a_r}\) are symmetric; the vectors \(c_{a_1},\ldots,c_{a_1\ldots a_{p-1}}\) with different combinations of indices are linearly independent; the rank of the collection of vectors \(c_{a_1},\ldots,c_{a_1\ldots a_p}\) is equal to \(n\), while otherwise the vectors \(c, c_{a_1\ldots a_r}\) are arbitrary.

1. Invariants

\[ \overset{\alpha}{H}_{a^1\ldots a^r}, \]

are constructed which possess the following property: every invariant embraced by the fundamental object of the surface is a function of them. This is done with the aid of the tensor

\[ a_{ab} \equiv T^{c^1c^2d^1\ldots d^{p-2}}_{ae^1e^2d^1\ldots d^{p-2}} T^{e^1e^2b^1\ldots b^{p-2}}_{bc^1c^2b^1\ldots b^{p-2}}; \]

of the tensor \(a^{ab}\), whose components are the reduced minors of the elements of the matrix \(\|a_{ab}\|\); of the tensors

\[ a^a \equiv a^{ac} T^{c^1c^2e^1\ldots e^{p-2}}_{cb^1b^2e^1\ldots e^{p-2}} a^{b^1b^2}a_{c^1c^2}, \qquad a^b_a \equiv T^{beb^1\ldots b^{p-2}}_{ac^1c^2b^1\ldots b^{p-2}} a^f a^{c^1c^2}a_{ef}; \]

of the tensors \(t^a_1,t^a_2,\ldots,t^a_m\), which are obtained as the result of the action of powers of the affinor \(a^b_a\) on the tensor \(a^a\), and of the tensors

\[ \Pi^\beta_{a_1\ldots a_r} \equiv T^\beta_{b_1\ldots b_r}t^{b^1}_{a_1}\cdots t^{b^r}_{a_r} \]

in the following way:

\[ \overset{\alpha}{H}_{a^1\ldots a^r} = \Pi^\beta_{a^1\ldots a^r}\Pi^\alpha_{\beta}, \qquad r=1,2,\ldots \]

Here \(\Pi^\alpha_{\beta}\) are the reduced minors of the elements of the matrix of components of the tensors
\[ \Pi^\alpha,\ldots,\Pi^\alpha_{a_1},\ldots,\Pi^\alpha_{a^1\ldots a^{p-1}} \]
and \(q\) tensors \(\Pi^\alpha_{\hat{\beta}}\) from among the tensors
\[ T^\alpha_{c^1\ldots c^p}M^{c^1\ldots c^p}_{b^1\ldots b^p}t^{b^1}_{a_1}\cdots t^{b^p}_{a_p}. \]
(The nondegeneracy of this matrix, and also of the matrix \(\|a_{ab}\|\), is verified in the case when \(q\) is either equal to \(m_p\) or is a divisor of \(m\).)

1. The vectors

\[ e_{a^1\ldots a^r} \equiv \Pi^\alpha_{a^1\ldots a^r}E_\alpha, \qquad r=1,2,\ldots,p-1, \qquad e_{\hat{\alpha}} \equiv \Pi^\beta_{\hat{\alpha}}E_\beta \]

are invariant. Their totality can be used as a canonical frame of the surface.

Received
20 XI 1957

Cited Literature

¹ G. F. Laptev, Tr. Moskovsk. matem. obshch., 2, 275 (1953).
² V. Hlavatý, Kon. Nederl. Akad. van Wetensh., Proc. 52, Nos. 5, 7, 9 (1949).
³ P. I. Shveikin, Tr. 3-go Vsesoyuzn. matem. s"ezda, 1, 1956, p. 175.