MATHEMATICS

N. I. KABANOV

ON THE GEOMETRIC THEORY OF THE SIMPLEST SINGULAR VARIATIONAL PROBLEM FOR AN \((n-1)\)-FOLD INTEGRAL

(Presented by Academician I. G. Petrovskii, 20 IV 1961)

Let us consider the variational problem for an \((n-1)\)-fold integral in parametric form:

\[ I=\int_{Q(t^a)}\cdots\int \Phi\left(\xi^\lambda,\xi_a^\lambda\right)\,dt^1\ldots dt^{n-1} \quad \left(\xi_a^\lambda=\frac{\partial \xi^\lambda}{\partial t^a}\right). \tag{1} \]

The requirement that the value of the integral (1) be invariant with respect to a transformation of the parameters,

\[ {}^{*}t^a=C^a(t^b) \quad \left(\operatorname{Det}|C_b^a|>0,\ C_b^a=\frac{\partial C^a}{\partial t^b}\right), \tag{2} \]

imposes a restriction on the integrand \(\Phi\):

\[ \Phi\left(\xi^\alpha,\xi_b^\alpha C_a^b\right) = \operatorname{Det}|C_a^b|\,\Phi\left(\xi^\alpha,\xi_c^\alpha\right). \tag{3} \]

It is easy to show that a function \(\Phi\) satisfying condition (3) has the form

\[ \Phi\left(\xi^\alpha,\xi_a^\lambda\right) = \mathfrak{H}\left(\xi^\lambda,\varepsilon_{\alpha\beta_1\ldots\beta_{n-1}} \xi_1^{\beta_1}\ldots \xi_{n-1}^{\beta_{n-1}}\right), \tag{4} \]

where the function \(\mathfrak{H}\) is positive and positively homogeneous of the first degree with respect to the second group of arguments, and \(\varepsilon_{\alpha\beta_1\ldots\beta_{n-1}}\) is the unit fundamental \(n\)-vector density of weight \(-1\). For the function (4), the integral (1) has the following form:

\[ \sigma=\int_{Q(t^a)}\cdots\int \mathfrak{H}\left(\xi^\lambda,\eta_\alpha\right)\,dt^1\ldots dt^{n-1} \quad \left(\eta_\alpha= \varepsilon_{\alpha\beta_1\ldots\beta_{n-1}} \xi_1^{\beta_1}\ldots \xi_{n-1}^{\beta_{n-1}}\right). \tag{5} \]

Interpreting the variables \(\xi^\alpha\) as coordinates of points in an \(n\)-dimensional geometric space \(X_n\), we shall call every hypersurface of the form

\[ \xi^\alpha=\xi^\alpha(t^a) \tag{6} \]

measurable if along it the integral (5) is meaningful. The value of the integral (5) for the hypersurface (6) is called its hyperarea\(^1\).

The concept of the indicatrix of the variational problem (5) is introduced essentially in the same way as for a variational problem expressed by an ordinary integral\(^2\). For fixed \(\xi^\lambda\), consider the equation

\[ \mathfrak{H}\left(\xi^\lambda,\eta_\alpha\right)=1, \tag{7} \]

which defines a hypersurface in the hyperplane coordinates \(x\) in the space \(\mathfrak{E}_n\) of all contravariant vector densities of weight \(+1\).

The variational problem (5) is called a singular variational problem of singularity class \(r\), if the rank of the matrix

\[ \left\| \mathfrak{H}^{\alpha\beta}\right\|,\qquad \left(\mathfrak{H}^{\alpha\beta}=\frac{\partial^2\mathfrak{H}}{\partial \eta_\alpha \partial \eta_\beta}\right) \tag{8} \]

is equal to \(n-1-r\). For \(r=0\) the variational problem (5) is called regular \((^1)\).

In the present note we shall be interested in the case when \(r=1\). Thus the indicatrices (7) will be hypersurfaces of the first class of singularity \((^2)\) in the composite manifold \((^3)\) \(\mathfrak{E}_n(X_n)\).

The remarks just made show that in considering the geometric theory of a singular variational problem for \((n-1)\)-fold integrals one must know the theory of a singular hypersurface of the 1st class of singularity in \(\mathfrak{E}_n\). Since \(\mathfrak{E}_n\), in a certain sense \((^1)\), is a centro-affine space, one may make use of the centro-affine theory of a singular hypersurface of the 1st class of singularity \((^2)\).

Assuming that the tangent hyperplanes of the singular hypersurface (7) do not pass through the center \(\mathfrak{E}_n\), its equation in hyperplane coordinates can be represented in parametric form:

\[ \eta_\alpha = l_\alpha(\eta^a)\qquad (a,b,\ldots=1,2,\ldots,n-2;\ \alpha,\ldots,\omega=1,2,\ldots,n). \tag{9} \]

The fundamental differential equations of the hypersurface (9) are written as follows \((^2)\):

\[ \nabla_a l_\alpha = l_{aa},\qquad \nabla_b l_{aa}=-h_{ba}l_\alpha-\mathfrak{G}_{ba}\mathfrak{R}_\alpha, \tag{10} \]

\[ \nabla_b \mathfrak{R}_\alpha =-\mathfrak{B}^{c}{}_{b}l_{\alpha c}-\mathfrak{B}_b l_\alpha, \tag{11} \]

where \(h_{ba}\) is a tensor, \(\mathfrak{G}_{ba}\), \(\mathfrak{B}^{c}{}_{b}\), \(\mathfrak{B}_a\) are \(\omega\)-densities of weights \(-\dfrac{2}{n-2}\) and \(\dfrac{2}{n-2}\). The \(n-1\) covariant vectors of the hypersurface \(l_\alpha\), \(l_{\alpha a}\) and the \(\omega\)-density \(\mathfrak{R}_\alpha\) of weight \(\dfrac{2}{n-2}\) are linearly independent. The covariant derivatives in (10) and (11) are taken with respect to the connection with coefficients

\[ G^c_{ba}=\mathfrak{G}^{cd}\mathfrak{n}^{\alpha}_{d}\partial_b l_{\alpha a}, \tag{12} \]

where \(\mathfrak{n}^{\alpha}_{a}\) is an \(\omega\)-density of weight \(-\dfrac{2}{n-2}\), satisfying the equations

\[ l_\alpha \mathfrak{n}^{\alpha}_{a}=0,\qquad l_{\alpha a}\mathfrak{n}^{\alpha}_{b}=\mathfrak{G}_{ab},\qquad \mathfrak{R}_\alpha \mathfrak{n}^{\alpha}_{a}=0, \tag{13} \]

and

\[ \mathfrak{G}_{ab}\mathfrak{G}^{bd}=\delta^d_a. \tag{14} \]

Suppose that for \(n>3\) the tensor \(h_{ba}\) does not degenerate,

\[ \mathfrak{h}=\operatorname{Det}|h_{ba}|\ne 0. \tag{15} \]

As is known \((^1)\), with each \(\mathfrak{E}_n\) one may associate a centro-affine space \(E_n\), and conversely; moreover, if densities in these spaces of covariant, contravariant valencies and weights respectively \(p,q,\mathfrak{k}\) and \(p,q,k\) are involved, then the equality holds:

\[ k=q-p-(n-1)\mathfrak{k}. \tag{16} \]

With the aid of the quantity

\[ \mathfrak{A}= \left( \frac{1}{(n-1)!}\, \mathfrak{E}^{\alpha\beta\alpha_1\ldots \alpha_{n-2}} \mathfrak{E}^{a_1\ldots a_{n-2}} l_\alpha \mathfrak{R}_\beta l_{\alpha_1 a_1}\cdots l_{\alpha_{n-2}a_{n-2}} \right)^2, \tag{17} \]

which is a scalar density of weight \(2\) in \(\mathfrak{E}_n\), and, according to (16), of weight \(-2(n-1)\) in \(E_n\) and of weight \(\dfrac{2n}{n-2}\) on the hypersurface, we shall put in corres—

quantities from \(\mathfrak{E}_n\) to the improper quantities from \(E_n\) (1) by means of the equalities

\[ \tilde l_\alpha = |\mathfrak h|^{\frac{n}{2(n-1)(n-2)}}\, \mathfrak A^{-\frac{1}{2(n-1)}}\,l_\alpha(\eta^a), \tag{18} \]

\[ {}^{*}\tilde l_{aa} = |\mathfrak h|^{\frac{n}{2(n-1)(n-2)}}\, \mathfrak A^{-\frac{1}{2(n-1)}}\,l_{aa}(\eta^b), \tag{19} \]

\[ \tilde N_\alpha = |\mathfrak h|^{-\frac{1}{2(n-1)}}\, \mathfrak A^{-\frac{1}{2(n-1)}}\, \mathfrak R_\alpha(\eta^b), \tag{20} \]

for which the derivational equations (10) and (11) take the form:

\[ \nabla_a \tilde l_\alpha = {}^{*}\tilde l_{aa} - \frac{n}{(n-1)(n-2)} A_a \tilde l_\alpha, \tag{21} \]

\[ \nabla_b {}^{*}\tilde l_{aa} = -\frac{n}{(n-1)(n-2)} A_b {}^{*}\tilde l_{aa} - h_{ba}\tilde l_\alpha - \tilde g_{ba}\tilde N_\alpha, \tag{22} \]

\[ \nabla_a \tilde N_\alpha = -\tilde V^c_{\cdot a}\,{}^{*}\tilde l_{ac} - \tilde W_a \tilde l_\alpha + \frac{1}{n-1} A_a \tilde N_\alpha, \tag{23} \]

where

\[ \tilde g_{ba} = |\mathfrak h|^{-\frac{1}{n-2}}\mathfrak G_{ba}, \tag{24} \]

\[ \tilde V^c_{\cdot a} = |\mathfrak h|^{-\frac{1}{n-2}}\mathfrak V^c_{\cdot a}, \tag{25} \]

\[ \tilde W_a = |\mathfrak h|^{-\frac{1}{n-2}}\mathfrak W_a, \tag{26} \]

\[ A_a=-\frac{1}{2}\nabla_a \ln |\mathfrak h|. \tag{27} \]

Alongside the tangent composite manifold of the first order \(E_n(X_n)\), consider the composite manifold \(\mathfrak E_n(X_n)\). The specification of the singular variational problem (5) entails the specification of a field of singular hypersurfaces in the composite manifold \(\mathfrak E_n(X_n)\). Assuming now that, in (9)—(27), all the quantities considered also depend on \(\xi^\alpha\), the equations of the field of singular hypersurfaces may be written in the form

\[ \eta_\alpha=l_\alpha(\xi^\lambda,\eta^a) \tag{28} \]

in hyperplane coordinates. Considering each hypersurface of the field (28) as \(X_{n-2}\), we arrive at consideration of the composite manifold \(X_{n+(n-2)}\). The principal task in studying this composite manifold is to find an invariant linear connection, determined by solving the Pfaff equations

\[ d\eta^a+\Gamma^a=0, \tag{29} \]

where \(\Gamma^a_\alpha(\xi^\lambda,\eta^b)\,d\xi^\alpha\) are the required Pfaff forms.

Corresponding to the quantities (18)—(20) and (28), introduce into consideration \(n\) independent Pfaff forms:

\[ \tilde l=\tilde l_\alpha(\xi^\lambda,\eta^b)\,d\xi^\alpha,\qquad {}^{*}\tilde l_a={}^{*}\tilde l_{aa}(\xi^\lambda,\eta^b)\,d\xi^\alpha,\qquad \tilde N=\tilde N_\alpha(\xi^\lambda,\eta^a)\,d\xi^\alpha . \tag{30} \]

Representing the operator of base differentiation \((^3)\) in the form of the symbolic equality

\[ D = {}^{*}\tilde l_a D^a + \tilde l\,D_{(n-1)} + \tilde N\,D_{(n)}, \tag{31} \]

and the required forms in the form of the expansion

\[ \Gamma^a = \gamma^{ab*}\tilde l_b + \gamma^a \tilde l + \delta^a \tilde N, \tag{32} \]

one can prove the following assertion.

If the indicatrices of the singular variational problem for an \((n-1)\)-fold integral are not cylindrical or conical surfaces, and their asymptotic cones are hypercones, while the tensor \(h_{ab}\) is nondegenerate, then the invariant linear connection in the composite manifold determined by the field of indicatrices, for \(n>3\), is uniquely determined from the conditions

\[ [\tilde l D\tilde l]=0,\qquad [\tilde N D\tilde l]=0,\qquad \underset{(n)}{D}\,\tilde g_{ab}=0, \tag{33} \]

where the square brackets denote the exterior product of forms in the sense of Cartan. For \(n=3\) the third condition is replaced by the condition

\[ \underset{(N)}{D}\,\mathfrak{W}^{(3)}=0, \tag{34} \]

where \(\mathfrak{W}^{(3)}\) is a scalar density of weight 3, replacing the \(w\)-density \(\mathfrak{W}_a\).

The linear connection in \(X_{n+(n-2)}\) found in this way can be used to find conditions for reducibility of the field of singular hypersurfaces in \(\mathfrak{E}_n(X_n)\) to a constant one, i.e. essentially the conditions for reducibility of the variational problem (5) to the case when the integrand contains no variables \(\xi^a\). The conditions found are indicated by the following

Theorem. In order that the field of singular hypersurfaces of the type under consideration in \(\mathfrak{E}_n(X_n)\) be reducible, by means of a suitable choice of coordinate systems, to a constant one, it is necessary and sufficient that, for \(n>3\), the linear connection in the composite manifold be a connection of zero curvature, that the fields of local objects \(G^c_{ab}, \tilde g_{ab}\), and \(h_{ab}\) be constant with respect to this connection, and that the scalar \(\varphi\) be identically equal to zero, where \(\varphi\) is the coefficient of the bracket \([\tilde l\tilde N]\) in the expansion of \([D\tilde l]\).

For \(n=3\) the indicated conditions are simplified.

Saratov State University
named after N. G. Chernyshevsky

Received
15 IV 1961

REFERENCES

¹ V. V. Wagner, Tr. seminara po vektorn. i tenzorn. analizu, 8, 144 (1950).
² V. V. Wagner, Matem. sborn., 21 (63), No. 3, 321 (1947).
³ V. V. Wagner, Tr. seminara po vektorn. i tenzorn. analizu, 8, 11 (1950).