Abstract
Full Text
UDC 513.013.3
MATHEMATICS
V. D. IZMAILOV
ON AN INVARIANT EQUIPPING OF A SURFACE OF A SPACE WITH AFFINE CONNECTION
(Presented by Academician A. D. Aleksandrov on 4 VI 1966)
An invariant equipping of a surface of a space with affine connection (with curvature and torsion) was considered earlier only for hypersurfaces \((^{1-3})\), for two-dimensional surfaces \((^4)\), and also for surfaces \(X_m\) of spaces \(L_n\), whose dimension \(n\) is equal to the dimension of the osculating space to \(X_m\) of some order \(p = 2, 3, 4, \ldots\) \((^5)\); G. F. Laptev solved this problem for \(n < m(m+3)/2\). In the present work a method is indicated for internally equipping a surface \(X_m\) of spaces \(L_n\) of arbitrary dimension.
Let \(X_m\) be given in \(L_n\), with connection object \(\Gamma_{jk}^{i}(x^l)\) \((i,j,k,l,\ldots = 1,2,\ldots,n)\), by the equations
\[ x^i = x^i(u^1,u^2,\ldots,u^m), \tag{1} \]
where the functions (1) and \(\Gamma_{jk}^{i}\) are assumed differentiable a sufficient number of times. As in \((^3,^4)\), put
\[ B_\alpha^{\,i} \equiv \partial_\alpha x^i,\qquad B_{\alpha\beta}^{\,i} \equiv \partial_\beta \partial_\alpha x^i \quad \text{and so on;} \]
\[ \bar B_{\alpha\beta}^{\,i} \equiv B_{\alpha\beta}^{\,i} + \Gamma_{(jk)}^{i} B_\alpha^{\,j} B_\beta^{\,k},\qquad \bar B_{\alpha\beta\sigma}^{\,i} \equiv \partial_\sigma \bar B_{\alpha\beta}^{\,i} + \Gamma_{(jk)}^{i} \bar B_{\alpha\beta}^{\,j} B_\sigma^{\,k} \quad \text{and so on} \]
\[ (\alpha,\beta,\sigma,\ldots = 1,2,\ldots,m). \]
\(B_\alpha^{\,i}\) is a tensor both in \(X_m\) and in \(L_n\); \(\bar B_{\alpha\beta}^{\,i}\) is a tensor only in \(L_n\). Let
\(T^{(1)}\{B_\alpha^{\,i}\}\), \(T^{(2)}\{\bar B_{\alpha\beta}^{\,i}, B_\alpha^{\,i}\}, \ldots,\)
\(T^{(p-1)}\{\bar B_{\alpha_1\cdots\alpha_{p-1}}^{\,i},\ldots,B_\alpha^{\,i}\}\) be the spaces tangent to \(X_m\) of orders \(1,2,\ldots,p-1\), where \(p\) is such a natural number that \(\dim T^{(p-1)} \le n\), while \(\dim T^{(p-1)}\) plus the number of vectors \(\bar B_{\alpha_1\cdots\alpha_p}^{\,i}\) is greater than \(n\). We shall assume that each \(T^{(q)}\) \((q \le p-1)\) has maximal dimension, i.e. that the vectors (relative to \(L_n\)) \(B_\alpha^{\,i},\ldots,\bar B_{\alpha_1,\ldots,\alpha_{p-1}}^{\,i}\) are linearly independent.
Consider the system of linear equations in \(m_i\)
\[ B_\alpha^{\,i} m_i = 0,\ldots,\bar B_{\alpha_1\cdots\alpha_{p-1}}^{\,i} m_i = 0. \tag{2} \]
It is invariant with respect to coordinate transformations both in \(X_m\) and in \(L_n\) and has \(N = n - \dim T^{(p-1)}\) linearly independent solutions \(m_i^{\,a}\) \((a,b,\ldots = 1,2,\ldots,N)\). Then
\[ \omega_{\alpha_1\cdots\alpha_p}^{\,a} \equiv \bar B_{\alpha_1\cdots\alpha_p}^{\,i} m_i^{\,a} \tag{3} \]
forms a system of \(N\) linearly independent tensors in \(X_m\).
If one chooses another system of solutions of (2),
\[ {}^{*}m_i^{\,b} = \lambda_a^{\,b} m_i^{\,a}\qquad \left(\det\left|\lambda_a^{\,b}\right| \ne 0\right), \]
then \(\omega_{a_1\ldots a_p}^{a}\) also undergoes the linear transformation
\[ {}^{*}\omega_{a_1\ldots a_p}^{b} = {}_{a}^{b}\lambda\,\omega_{a_1\ldots a_p}^{a}, \tag{4} \]
i.e., the index \(a\) for \(\omega_{a_1\ldots a_p}^{a}\) has, with respect to the transformations (4), a tensorial character.
Put
\[ A^{a_1\ldots a_m} \equiv \omega_{\underline{a_1\ldots a_p}}^{a_1}\cdots \omega_{\underline{a_1^m\ldots a_p^m}}^{a_m}, \tag{5} \]
where alternation is performed over the indices underlined by identical signs. Obviously, \(A^{a_1\ldots a_m}\) \((a_\alpha=1,2,\ldots,N)\) is a tensor with respect to the transformations (4) (a \(\lambda\)-tensor) and a scalar density of weight \(p\) in \(X_m\). We shall now distinguish two cases.
I. If \(p\) is even, then \(A^{a_1\ldots a_m}\) is symmetric. We shall assume that it has nonzero discriminant. Then, as follows from the works \((^{7-10})\), there exists a \(\lambda\)-tensor \(\widetilde A_{a_1\ldots a_m}\) such that
\[
\widetilde A_{a a_2\ldots a_m}A^{b a_2\ldots a_m}=\delta_a^b.
\]
In \(X_m\) it is a tensor density of weight \(-p\).* We now define in \(X_m\) a symmetric tensor density of weight \(-p\):
\[ \mathfrak H_{a_1'\ldots a_p'\ldots a_1^m\ldots a_p^m} \equiv \omega_{(a_1'\ldots a_p')}^{a_1}\cdots \omega_{(a_1^m\ldots a_p^m)}^{a_m} \widetilde A_{a_1\ldots a_m}. \tag{6} \]
Assuming the discriminant of this density to be nonzero and dividing it by a suitable power of its discriminant, we obtain a “pure” tensor
\[
H_{a_1'\ldots a_p'\ldots a_1^m\ldots a_p^m}.
\]
In this case, by the method of Lieber—Ermakov—Kramlet (see \((^{7-9})\)), we can invariantly associate with \(H_{a_1'\ldots a_p'\ldots a_1^m\ldots a_p^m}\) a connection \(\widetilde\Gamma_{\alpha\beta}^{\sigma}\), which is invariant in \(L_n\).
II. If \(p\) is odd, then \(A^{a_1\ldots a_m}\) is skew-symmetric. Consider the system of linear homogeneous equations with respect to \(C^{\alpha_1\ldots\alpha_p\alpha_{p+1}}\):
\[ C^{\alpha_1\ldots\alpha_p\alpha_{p+1}}\omega_{\alpha_1\ldots\alpha_p}^{a}=0 \quad (\alpha_1\ldots\alpha_{p+1}=1,\ldots,m). \tag{7} \]
It is easy to calculate that the number \(k\) of equations in this system and the number \(l\) of unknowns are connected by the condition \(k\le l\). Indeed, \(l\) is equal to the number of combinations with repetitions from \(m\) taken \(p\) at a time, multiplied by \(m\): \(l=\widetilde C_m^p m\); \(k=Nm=(n-\dim T^{(p-1)})m\le(\)the number of vectors \(B_{\alpha_1\ldots\alpha_p}^{i}+\dim T^{(p-1)}-\dim T^{(p-1)}\) \()m\), or \(k\le\widetilde C_m^p m\), i.e. \(k\le l\). Equality is attained only in the case \(n=\dim T^{(p)}\), for which the connection in \(X_m\) is introduced by the method of Hlavatý (see \((^{5})\)). Excluding this case, we have \(k<l\), and the system (7) has a certain number \(K\) of linearly independent solutions
\[
C_u^{\alpha_1\ldots\alpha_p\alpha_{p+1}}
\quad
(v,u=1,2,\ldots,K).
\]
These tensors** are determined up to a linear transformation:
\[ {}^{*}C_{u}^{\alpha_1\ldots\alpha_p\alpha_{p+1}} = {}_{u}^{v}\Lambda\, C_{v}^{\alpha_1\ldots\alpha_p\alpha_{p+1}}. \tag{8} \]
* To form a pseudotensor “reciprocal” to a given pseudotensor, one may use any of its relative invariants; for example, its multidimensional determinant is convenient (see \((^{10})\)).
** The tensorial character of the quantities \(C_u^{\alpha}\) is established directly.
Again we shall say that \(C_u^a\cdots\) relative to \(u\) is a \(\Lambda\)-tensor. Analogously to (5), introduce the \(\Lambda\)-tensor
\[ B_{u_1\ldots u_m}\equiv C_{u_1}^{\hat{\alpha}_1\ldots \hat{\alpha}_{p+1}}\cdots C_{u_m}^{\hat{\beta}_1\ldots \hat{\beta}_{p+1}}, \tag{9} \]
where the hats above again indicate alternation. In this case \(B_{u_1\ldots u_m}\) will be symmetric. Assuming the discriminant of (9) to be nonzero, we construct for it the reciprocal \(\Lambda\)-tensor \(\widetilde B^{u_1\ldots u_m}\), and then the tensor density in \(\dot X_m\)
\[ \mathfrak h^{\alpha_1'\ldots \alpha_p'\ldots \alpha_1^m\ldots \alpha_p^m} \equiv C_{u_1}^{(\alpha_1'\ldots \alpha_p'}\cdots C_{u_m}^{\alpha_1^m\ldots \alpha_p^m)} \widetilde B^{u_1\ldots u_m}, \tag{10} \]
which is again sufficient for introducing a connection \(\widetilde\Gamma_{\alpha\beta}^{\sigma}\) in \(\dot X_m\).
We note that the differentiability classes of the functions \(\Gamma^i_{jk}\) and (1), used for constructing \(\widetilde\Gamma_{\alpha\beta}^{\sigma}\), are respectively equal to \(C^{p-1}\) and \(C^{p+1}\). For \(X_m\) in \(L_n\), where \(n<m(m+3)/2\), we obtain \(C^1\) and \(C^3\).
Introduce the following system of quantities:
\[ \widetilde\omega_{a}^{\sigma_1\ldots\sigma_p} \equiv \frac1{m!}\, \omega_{a_1'\ldots\alpha_p'}^{a_1}\cdots \omega_{\alpha_1^{m-1}\ldots\alpha_p^{m-1}}^{a_{m-1}} A_{\alpha_1\ldots\alpha_{m-1}a}\, \delta_{1\ldots m}^{\alpha_1'\ldots\alpha_1^{m-1}\sigma_1} \cdots \delta_{1\ldots m}^{\alpha_p'\ldots\alpha_p^{m-1}\sigma_p}, \]
where \(\delta_{1\ldots m}^{\alpha_1'\ldots\alpha_1^{m-1}\sigma_1}\) (the indices range from 1 to \(m\)) is the generalized Kronecker symbol (contraction of a tensor with \(\frac1{m!}\delta_{1\ldots m}^{\alpha_1\ldots\alpha_m}\) is equivalent to its alternation with respect to the indices \(\alpha_1\ldots\alpha_m\)). It is easy to see that
\[ \widetilde\omega_{a}^{\sigma_1\ldots\sigma_p} \omega_{\sigma_1\ldots\sigma_p}^{b} = \delta_a^b \qquad (a,b=1,\ldots,N). \tag{11} \]
We define the invariant equipment by the spaces
\[ \mathfrak P^{(2)}\{D_\beta B_a^i\},\quad \mathfrak P^{(3)}\{D_\beta D_\gamma B_\delta^i\},\ldots,\quad \mathfrak P^{(p-1)}\{D_{\sigma_1}\cdots D_{\sigma_{p-2}}B_{\sigma_{p-1}}^i\} \]
(spanned by the vectors indicated in braces) and by the system of vectors
\[ n_a^i \equiv \widetilde\omega^{\sigma_1\ldots\sigma_p} D_{\sigma_1}\cdots D_{\sigma_{p-1}}B_{\sigma_p}^i . \]
Here \(D_\sigma\) is the operator of mixed differentiation with respect to the pair of connections \(\Gamma^i_{jk}\), \(\widetilde\Gamma_{\alpha\beta}^{\sigma}\), where \(\widetilde\Gamma_{\alpha\beta}^{\sigma}\) has been introduced by the method described above. It is easy to see that, by virtue of the assumptions made at the beginning of the paper and of (11), we obtain a genuinely invariant equipment. The connection \(\Gamma_{\alpha\beta}^{\sigma}\) induced by this equipment, generally speaking, does not coincide with \(\widetilde\Gamma_{\alpha\beta}^{\sigma}\).
Sverdlovsk State
Pedagogical Institute
Received
26 V 1966
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