## Some new obstructions to minimal and Lagrangian isometric immersions.(English)Zbl 1026.53009

The author presents a refinement of his $$\delta$$-invariant for isometric immersions. Let $$M$$ be an $$n$$-dimensional Riemannian manifold with sectional curvature $$K$$. For every subspace $$L\subset T_pM$$ with $$\dim L\geq 2$$ he defines $$\tau(L):= \sum_{i<j} K(e_i\wedge e_j)$$, where $$e_1,\dots,e_r$$ denotes an orthonormal basis of $$L$$. In particular, $$\tau(T_pM)$$ is the scalar curvature of $$M$$ at $$p$$. Furthermore he defines $\delta(n_1, \dots,n_k) (p):= \tau(T_pM)- \inf\left\{ \sum^k_{i=1} \tau(L_i)\mid L_i\subset T_pM,\;\dim L_i= n_i,\;L_i\perp L_j\right\}$ (for each family $$(n_1,\dots,n_k)$$ for which this definition makes sense) and numbers $$b(n_1,\dots,n_k)\in\mathbb{Z}$$ and $$c(n_1,\dots, n_k)\in \mathbb{Q}$$ (only depending on $$(n_1,\dots,n_k))$$. The main result is the following theorem: If $$M$$ is a submanifold of a real space form $$R^m( \varepsilon)$$ of constant curvature $$\varepsilon$$ and $$H$$ denotes the mean curvature of $$M$$, then the inequality $\delta(n_1,\dots,n_k)\leq c(n_1,\dots, n_k)\cdot H^2 +b(n_1,\dots,n_k) \cdot\varepsilon$ is always true, and equality holds if and only if the shape operator of $$M$$ is of a special kind. The same result is true for totally real submanifolds in complex space forms of constant, holomorphic sectional curvature $$4\varepsilon$$. The curvature invariants $$\delta(2,\dots,2)$$ are of particular interest. Several applications are described.

### MSC:

 53B25 Local submanifolds 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)