[Paper Review] Invariant, anti-invariant and slant submanifolds of a metallic Riemannian manifold

Adara M. Blaga, Cristina E. Hreţcanu|arXiv (Cornell University)|Mar 4, 2018

Geometric Analysis and Curvature Flows5 references15 citations

TL;DR

This paper investigates invariant, anti-invariant, and slant submanifolds within metallic Riemannian manifolds, introducing a canonical $Σ$-structure induced by the metallic structure $J$. It proves that the product of spheres $S^{a-1}(r_1) \times S^{b-1}(r_2)$ inherits a metallic Riemannian structure via the induced tensor $T$, confirming it as an invariant submanifold when isometrically embedded in $E^{a+b}$, with explicit formulas for the normal components of $J$ and the induced structure's properties.

ABSTRACT

Properties of invariant, anti-invariant and slant isometrically immersed submanifolds of metallic Riemannian manifolds are given with a special view towards the induced $Σ$-structure. Examples of such metallic manifolds are also given.

Motivation & Objective

• To characterize invariant, anti-invariant, and slant submanifolds in metallic Riemannian manifolds.
• To study the inheritance of metallic structures on submanifolds via the induced $\Sigma$-structure.
• To establish geometric and algebraic properties of the induced metallic structure on isometrically immersed submanifolds.
• To provide explicit examples of metallic Riemannian manifolds, including the product of spheres $S^{a-1}(r_1) \times S^{b-1}(r_2)$.
• To analyze the behavior of the metallic structure $J$ under decomposition into tangential and normal components on submanifolds.

Proposed method

• Utilizes the Gauss and Weingarten equations to decompose the metallic structure $J$ into tangential ($T$) and normal ($N$) components on submanifolds.
• Defines the induced structure $(T, \langle\cdot,\cdot\rangle, 0, 0, 0, 0, \mathcal{A})$ on the submanifold, where $\mathcal{A}$ encodes the normal components of $J$.
• Applies the generalized secondary Fibonacci sequence $\{g_n\}$ to express powers of $J$ as $J^n = g_n J + q g_{n-1} I_{\Gamma(TM)}$.
• Constructs explicit examples by embedding $S^{a-1}(r_1) \times S^{b-1}(r_2)$ into $E^{a+b}$ and computing $J$-action on tangent and normal vectors.
• Derives the matrix $\mathcal{A} = \frac{1}{r^2}\begin{pmatrix} \sigma r_1^2 + \overline{\sigma} r_2^2 & r_1 r_2 (\sigma - \overline{\sigma}) \\ r_1 r_2 (\sigma - \overline{\sigma}) & \overline{\sigma} r_1^2 + \sigma r_2^2 \end{pmatrix}$ for the normal component of $J$.
• Verifies that $J$ preserves the normal bundle by showing $J(T_x^\perp M) \subseteq T_x^\perp M$, implying the induced $T$ satisfies the metallic equation $T^2 = pT + qI$.

Experimental results

Research questions

• RQ1 How does a metallic Riemannian structure on an ambient manifold induce a structure on its submanifolds?
• RQ2 Under what conditions is a submanifold invariant, anti-invariant, or slant with respect to the metallic structure $J$?
• RQ3 What is the form of the induced metallic structure $T$ on a submanifold, and how is it related to the ambient $J$?
• RQ4 Can explicit examples of metallic Riemannian manifolds be constructed, such as products of spheres?
• RQ5 How do the normal components of $J$ on a submanifold relate to the induced structure and the ambient geometry?

Key findings

• The product of spheres $S^{a-1}(r_1) \times S^{b-1}(r_2)$ is an invariant submanifold of $E^{a+b}$ under the metallic structure $J$, as $J(T_x M) \subseteq T_x M$.
• The induced tangential component $T$ satisfies $T^2 = pT + qI$, confirming it is a metallic structure on the submanifold.
• The normal components of $J$ are encoded in the matrix $\mathcal{A}$, with entries $a_{11} = \frac{\sigma r_1^2 + \overline{\sigma} r_2^2}{r^2}$, $a_{12} = a_{21} = \frac{r_1 r_2 (\sigma - \overline{\sigma})}{r^2}$, and $a_{22} = \frac{\overline{\sigma} r_1^2 + \sigma r_2^2}{r^2}$.
• The tangential and normal projections of $J$ satisfy $g(TX, Y) = g(X, TY)$ and $g(nU, V) = g(U, nV)$, confirming symmetry of $T$ and $n$ with respect to the metric.
• The normal bundle is preserved under $J$, as $J(T_x^\perp M) \subseteq T_x^\perp M$, which follows from $\xi_1 = \xi_2 = 0$ in the decomposition $JN_k = \xi_k + \sum a_{k\alpha} N_\alpha$.
• The induced structure on $S^{a-1}(r_1) \times S^{b-1}(r_2)$ is a metallic Riemannian structure $(T, \langle\cdot,\cdot\rangle)$, with the ambient metric $\langle\cdot,\cdot\rangle$ compatible with $J$.