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Strings of Riemannian invariants, inequalities, ideal immersions and their applications. (English) Zbl 1009.53041
Choe, Jaigyoung (ed.), The third Pacific Rim geometry conference. Proceedings of the conference, Seoul, Korea, December 16-19, 1996. Cambridge, MA: International Press. Monogr. Geom. Topology. 25, 7-60 (1998).
Let $$M$$ be an $$n$$-dimensional submanifold of a real space form $$\widetilde M(c)$$ with constant sectional curvature $$c$$. In [Arch. Math. 60, No. 6, 568-578 (1993; Zbl 0811.53060)] the author proved that the Riemannian invariant $$\delta_M$$ of $$M$$, defined by $$\delta_M(p)=\tau(p)-\inf K(p)$$, where $$\inf K(p)$$ is the smallest sectional curvature at $$p\in M$$ and $$\tau=\sum_{i<j}K(e_i\wedge e_j)$$ is the scalar curvature, satisfies the following inequality: $\delta_M\leq{n^2(n-2)\over 2(n-1)} |H|^2+\frac 12 (n+1)(n-2)c,$ where $$H$$ is the mean curvature vector. In [Result. Math. 27, 17-26 (1995; Zbl 0834.53045)] the author considered immersions of a compact Riemannian $$n$$-manifold into Kählerian $$n$$-manifolds with constant holomorphic sectional curvature. An immersion $$f: M\to \widetilde M$$ from a Riemannian $$n$$-manifold $$M$$ into a Hermitian manifold $$\widetilde M$$ is called Lagrangian if the complex structure $$J$$ of $$\widetilde M$$ maps each tangent space of $$M$$ into its normal space and $$\dim M=\dim_{\mathbb{C}}\widetilde M$$. More generally, $$f: M\to \widetilde M$$ is said to be slant if its slant angle is constant, i.e. $$\angle (J(f_*X),f_*(T_pM))=\alpha$$ for all $$X\in T_pM-\{0\}$$ and $$p\in M$$, where $$\angle$$ denotes the angle in $$\widetilde M$$ with respect to the metric of $$\widetilde M$$. The author proved that a compact Riemannian $$n$$-manifold with finite fundamental group satisfying $$\delta_M>0$$ admits no slant immersion into any flat Kählerian $$n$$-manifold $$\widetilde M$$. In particular, $$M$$ admits no Lagrangian immersion into any complex $$n$$-torus $$\mathbb{C} T^n$$.
In this paper the author studies the two strings of new type of Riemannian curvature invariants and the sharp inequalities, involving these invariants and the squared mean curvature of a Riemannian manifold $$M$$. For given integers $$n\geq 2$$ and $$k \geq 0$$, let $${\mathcal S}(n)= \{(n_1,\dots, n_k)\in\mathbb{Z}^k: 2 \leq n_1 \leq \dots \leq n_k < n\}$$. For a subspace $$L \subset T_pM$$, $$p\in M$$, denote $$\tau(L)=\sum_{i<j}K(e_i\wedge e_j)$$, where $$\{e_1,\dots,e_k\}$$ is a basis of $$L$$ and $$K$$ is the sectional curvature of $$M$$. The generalized curvature invariants are defined as follows: \begin{aligned} \delta(n_1,\dots,n_k)&= \tau(T_pM) - \inf\{\tau(L_1)+\dots+\tau(L_k)\}\\ \text{and}\widetilde\delta(n_1,\dots,n_k)&= \tau(T_pM)- \sup\{\tau(L_1)+\dots+\tau(L_k)\}, \end{aligned} where $$L_1,\dots, L_k$$ run over all $$k$$ mutually orthogonal subspaces of $$T_pM$$ such that dim $$L_j=n_j$$, $$1\leq j \leq k$$.
Any submanifold $$M$$ of a space form with constant sectional curvature $$\varepsilon$$ satisfies the sharp inequalities $\delta(n_1,\dots,n_k)\leq c(n_1,\dots,n_k)H^2 + b(n_1,\dots,n_k)\epsilon,$ for all $$(n_1,\dots,n_k) \in {\mathcal S}(n)$$, where $$b(n_1,\dots,n_k)$$ and $$c(n_1,\dots,n_k)$$ are suitable constants. The invariants $$\delta(n_1,\dots,n_k)$$ and $$\widetilde\delta(n_1,\dots$$, $$n_k)$$ have several interesting connections to several areas of mathematics. For instance, they give rise to new obstructions to minimal, Lagrangian and slant isometric immersions. Moreover, they relate closely to the notion of order and to the first nonzero eigenvalue $$\lambda_1$$ of the Laplacian $$\Delta$$ on a Riemannian manifold, and together with the sharp inequalities give rise naturally to the notion of ‘ideal immersions’ or the notion of ‘the best ways of living’.
Many examples of ideal immersions are presented in this paper. The author also gives classification, existence and non-existence theorems, and explains the method for applying the sharp inequalities to establish rigidity theorems for submanifolds in space forms without global assumption on the submanifolds and regardless of codimension. Finally, the author presents the sharp relationships between the $$k$$-Ricci curvatures, squared mean curvature, and the shape operator for an arbitrary submanifold in a Riemannian space form.
For the entire collection see [Zbl 0953.00035].

##### MSC:
 53C40 Global submanifolds 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53C20 Global Riemannian geometry, including pinching 53D12 Lagrangian submanifolds; Maslov index