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On the DDVV conjecture and the comass in calibrated geometry. I. (English) Zbl 1180.53055
Let $$M^n$$ be a $$n$$-dimensional manifold isometrically immersed into the space form $$N^{n+m}(c)$$ of constant sectional curvature $$c$$. In the study of submanifolds theory, De Smet, Dillen, Verstraelen and Vrancken made the following (DDVV) conjecture: Let $$h$$ be the second fundamental form, let $$H=\frac 1n\text{trace}\,h$$ be the mean curvature tensor, let $$\rho$$ and $$\rho^\perp$$ be the normalized scalar curvature for the tangent and the normal bundle, resp. Then $$\rho+\rho^\perp\leq | H|^2+C$$. This Conjecture, in terms of matrices, can be reformulated as follows conjecture $$P(n,m)$$. Let $$A_1,\dots,A_m$$ be the traceless $$n\times n$$ symmetric matrices. For $$n,m\geq 2$$ we have $\left(\sum^m_{r=1}\| A_r\|^2\right)^2\geq 2\left(\sum_{r<s} \|[A_r,A_s]\|^2\right).$ The main result of the paper is the following Theorem. Let $$A_i(i=1,\dots,m)$$ be $$3\times 3$$ symmetric matrices. Then $\left(\sum^m_{i=1}\| A_i\|^2\right)^2\geq 2 \left(\sum_{i<j}\| [A_i,A_js]\|^2\right).$ That is, the Conjecture $$P(3,m)$$ for $$m\geq 2$$ is true.

##### MSC:
 53C38 Calibrations and calibrated geometries 53C40 Global submanifolds
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##### References:
 [1] Bryant R.L. (1991). Some remarks on the geometry of austere manifolds. Bol. Soc. Brasil. Mat. (N.S.) 21(2): 133–157 · Zbl 0760.53034 [2] Chen B.-Y. (1996). Mean curvature and shape operator of isometric immersions in real-space-forms. Glasgow Math. J. 38(1): 87–97 · Zbl 0866.53038 [3] Chern, S.S., do Carmo, M., Kobayashi, S.: Minimal submanifolds of a sphere with second fundamental form of constant length. In Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968), pp. 59–75. Springer, New York (1970) · Zbl 0216.44001 [4] Dadok J. and Harvey R. (1999). The Pontryagin 4-form. Proc. Am. Math. Soc. 127(11): 3175–3180 · Zbl 0943.53035 [5] Dajczer M. and Florit L.A. (2001). A class of austere submanifolds. Illinois J. Math. 45(3): 735–755 · Zbl 0988.53004 [6] De Smet P.J., Dillen F., Verstraelen L. and Vrancken L. (1999). A pointwise inequality in submanifold theory. Arch. Math. (Brno) 35(2): 115–128 · Zbl 1054.53075 [7] Dillen, F., Fastenakels, J., Veken, J.: Remarks on an inequality involving the normal scalar curvature. DG/0610721 (2006) · Zbl 1143.53060 [8] Gluck H., Mackenzie D. and Morgan F. (1995). Volume-minimizing cycles in Grassmann manifolds. Duke Math. J. 79(2): 335–404 · Zbl 0837.53035 [9] Gu W. (1998). The stable 4-dimensional geometry of the real Grassmann manifolds. Duke Math. J. 93(1): 155–178 · Zbl 0943.53036 [10] Harvey R. and Lawson H.B. (1982). Calibrated geometries. Acta Math. 148: 47–157 · Zbl 0584.53021 [11] Lu, Z.: Proof of the normal scalar curvature conjecture, preprint [12] Suceavă, B.: Some remarks on B. Y. Chen’s inequality involving classical invariants. An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.). 45(2), 405–412 (2000), 1999 · Zbl 1011.53042 [13] Suceavă, B.D.: DDVV conjecture. preprint
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