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), 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)).}^{\alpha_1},\ldots,\Lambda_{a^1,\ldots,a^w}^{\alpha

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),}^{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}\equiv}^{b^1\ldots b^s,c^1\ldots c^{r+1-s}
\mathcal{B}{(a\ldots a^s}^{b^1\ldots b^s}\mathcal{B});}\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}) is the unit (m)-vector;
(\mathcal{E}{a_1^1\ldots a_1^r,\ a_2^1\ldots a_2^r,\ldots,\ a) is a relative tensor, symmetric in the indices belonging to one series and skew-symmetric with respect to the series, where}^1\ldots a_{m_r}^r
[
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,}^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) the formula has been obtained}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

[
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_b,}\mathfrak{B
\qquad
\mathfrak{K}^{a,b^1\ldots b^s}{a^1\ldots a^{s+r},b}
=
\mathfrak{B}^{b^1\ldots b^s}},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},
]

[
m^{b^1\ldots b^s}{a^1\ldots a^{s+r}}
=
-\sum\,}^{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}\,},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}^{p
\mathop{m}^{*\,bc^1\ldots c^{p-r-1}}{d^1\ldots d^s}
\left(
M^{d^1\ldots d^s}}c^1\ldots c^{p-r-1}
+
M^{d^1\ldots d^s}{e^1\ldots e^p}
m^{e^1\ldots e^p}}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}^{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}}
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},
\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}
=
\mathfrak{B}^{b^1\ldots b^s}{a^1\ldots a^t}\mathfrak{B}^{a},
]

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

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

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

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}), 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.
]

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}
\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},
\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}^{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}};}
]

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,
\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;
]

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}
]

in the following way:

[
\overset{\alpha}{H}{a^1\ldots a^r}
=
\Pi^\beta,}\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}}\cdots t^{b^p{a_p}.
]
(The nondegeneracy of this matrix, and also of the matrix (|a|), 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.