A unimodularly invariant theory for immersions into the affine space (Q1849684)

The paper deals with the framing of a \(C^\infty\)-immersion \(x\) of a \(C^\infty\)-manifold \(M^n\) into the affine space \(A^{n+p}\) as this has been done by \textit{C. Burstin} and \textit{W. Mayer} [Math. Z. 27, 373-407 (1927; JFM 53.0707.06)], \textit{K. H. Weise} [Math. Z. 44, 161-184 (1938; Zbl 0019.18501)] and \textit{W. Klingenberg} [Math. Z. 54, 65-80 (1951; Zbl 0044.17801)]. Its aim is to give a new unimodular theory for higher codimension \(p\) by generalization of Blaschke's theory of hypersurfaces. The first of the author's principal ideas is the introduction of a \((0,2p)\)-tensor field \(g\) on \(M^n\) and its volume form \(\omega(g): =|\det (g)|^{1\over 2p}\) by the formulas \[ \Lambda (u_1,v_1, \dots,u_p,v_p): =\overline {\text{Det}}\bigl(dx (\partial_1), \dots,dx (\partial_n), \overline\nabla_{u_1} \overline\nabla_{v_1}x, \dots, \overline \nabla_{u_p} \overline\nabla_{v_p} x\bigr) \] and \(f:= \Lambda\cdot |\det (\Lambda)|^{-{1 \over n+2p}}\) with a suitable chosen generalized determinant \(\det(\cdot)\) for regular immersions \(x(\omega (g)\neq 0)\). As second idea the choice of ``unimodular'' transversal spaces \(\sigma(u): =\text{span} \{y_1,\dots, y_p\}\) of \(x\) may be regarded which fulfil a generalized apolarity condition and additionally a normalization condition. Then the main result is the characterization of a critical point \(x\) of the unimodular area functional \(A(M^n): =\int_{M^n} \omega(g)\) by the vanishing of its mean curvature vector relative to \(\sigma (u)\). Finally it should be mentioned that the Ricci tensor of the connection \(\nabla\) on \(M^n\), induced by \(\sigma(u)\), is symmetric.