[Paper Review] Hemi-slant submanifolds in metallic Riemannian manifolds

Cristina E. Hreţcanu, Adara M. Blaga|arXiv (Cornell University)|Apr 14, 2018

Geometric Analysis and Curvature Flows16 references3 citations

TL;DR

This paper investigates hemi-slant submanifolds in metallic and Golden Riemannian manifolds, establishing integrability conditions for the distributions $D^\theta$ and $D^\perp$ via curvature and connection analysis. Key results include necessary and sufficient conditions for $D^\perp$ integrability and mixed totally geodesic submanifolds, with explicit eigenvalue formulas for the shape operator when $\overline{\nabla}_X N Y = 0$. The work generalizes slant geometry to metallic structures using the polynomial equation $J^2 = pJ + qI$, extending prior results on semi-slant and pseudo-slant submanifolds.

ABSTRACT

The aim of our paper is to focus on some properties of hemi-slant submanifolds in metallic (and Golden) Riemannian manifolds. We give some characterizations for submanifolds to be hemi-slant submanifolds in metallic (or Golden) Riemannian manifolds and we obtain integrability conditions for the distributions involved. Examples of hemi-slant submanifolds of metallic and Golden Riemannian manifolds are given.

Motivation & Objective

• To investigate the geometric properties of hemi-slant submanifolds in metallic and Golden Riemannian manifolds.
• To derive integrability conditions for the distributions $D^\theta$ and $D^\perp$ associated with the submanifold decomposition.
• To characterize conditions under which hemi-slant submanifolds are $D^\theta$-$D^\perp$ mixed totally geodesic.
• To extend known results on slant and semi-slant submanifolds to the broader class of metallic structures.

Proposed method

• The paper uses the metallic structure defined by $J^2 = pJ + qI$ for $p,q \in \mathbb{N}^*$, generalizing the Golden structure when $p=q=1$.
• It decomposes the tangent and normal bundles of the submanifold $M$ into $D^\theta$ (slant distribution) and $D^\perp$ (anti-invariant distribution), analyzing the induced tensor fields $T$, $N$, $t$, and $n$.
• The integrability of $D^\perp$ is analyzed via the condition $(\nabla_Z T)W = (\nabla_W T)Z$ for $Z,W \in \Gamma(D^\perp)$, linking it to the vanishing of the shape operator $A_{NZ}W$.
• The mixed totally geodesic condition is characterized by requiring $A_V X \in \Gamma(D^\theta)$ and $A_V Y \in \Gamma(D^\perp)$ for $X \in \Gamma(D^\theta)$, $Y \in \Gamma(D^\perp)$, and $V \in \Gamma(T^\perp M)$.
• The paper derives eigenvalue formulas for the shape operator $n$ acting on $h(X,Y)$ when $\overline{\nabla}_X N Y = 0$, solving $\lambda^2 - p\cos^2\theta \lambda - q\cos^2\theta = 0$.
• Examples of hemi-slant submanifolds are constructed in metallic and Golden Riemannian manifolds to illustrate the theoretical framework.

Experimental results

Research questions

• RQ1 Under what conditions is the anti-invariant distribution $D^\perp$ integrable in a hemi-slant submanifold of a metallic Riemannian manifold?
• RQ2 When is a hemi-slant submanifold $D^\theta$-$D^\perp$ mixed totally geodesic, and what conditions ensure this property?
• RQ3 What are the eigenvalues of the shape operator $n$ acting on the second fundamental form $h(X,Y)$ when $\overline{\nabla}_X N Y = 0$ for $X,Y \in \Gamma(D^\theta)$?
• RQ4 How do the curvature and connection properties of the ambient metallic manifold constrain the geometry of the submanifold's distributions?

Key findings

• The anti-invariant distribution $D^\perp$ is integrable if and only if $(\nabla_Z T)W = (\nabla_W T)Z$ for all $Z,W \in \Gamma(D^\perp)$, establishing a connection between the covariant derivative of $T$ and the integrability of $D^\perp$.
• If $A_{NZ}W = 0$ for all $Z,W \in \Gamma(D^\perp)$, then $D^\perp$ is integrable, providing a sufficient condition based on the shape operator.
• When $\overline{\nabla}_X N Y = 0$ for all $X,Y \in \Gamma(D^\theta)$, the second fundamental form $h(X,Y)$ is either zero (geodesic) or an eigenvector of $n$ with eigenvalues $\lambda_{1,2} = \frac{p\cos^2\theta \pm \cos\theta \sqrt{p^2\cos^2\theta + 4q}}{2}$.
• A hemi-slant submanifold is $D^\theta$-$D^\perp$ mixed totally geodesic if and only if $A_V X \in \Gamma(D^\theta)$ and $A_V Y \in \Gamma(D^\perp)$ for all $X \in \Gamma(D^\theta)$, $Y \in \Gamma(D^\perp)$, and $V \in \Gamma(T^\perp M)$.
• If $\overline{\nabla}_X N Z = 0$ for all $X \in \Gamma(TM)$, $Z \in \Gamma(D^\perp)$, then $h(X,Z) = 0$ for all $X \in \Gamma(D^\theta)$, $Z \in \Gamma(D^\perp)$, implying $D^\theta$-$D^\perp$ mixed total geodesy.
• The paper provides explicit examples of hemi-slant submanifolds in metallic and Golden Riemannian manifolds, demonstrating the realizability of the theoretical framework.