UC Riverside

\indent In 1997, Wilhelm \cite{OnIntR} proved the following generalization of Synge's Theorem: let $(M, g_M)$ be a compact Riemannian $n$-manifold with $\textnormal{Ric}_k (M, g_M)\geq k$ and $\textnormal{sys}_1(M, g_M)> \pi\sqrt{\frac{k-1}{k}}$; if $n$ is even and $M$ is orientable, then $M$ is simply connected; if $n$ is odd, then $M$ is orientable (Theorem \ref{Fredsmainresult}). Furthermore, he proved that this lower bound on $\textnormal{sys}_1$ is optimal when $k=n-1$. In 2020, Mouillé \cite{LawrenceThesis} proved that $S^3\times S^3$ admits a metric $g_\ell$ with $\textnormal{Ric}_2(S^3\times S^3, g_\ell)>0$ (Theorem \ref{LawrenceMain}). \vskip 1em

In this dissertation, we first show that the metric $g_\ell$ (which is a Cheeger deformation) is canonical variation. This follows from a more general result we prove (Theorem \ref{nhmetric}), which is that if $(M, g_\ell)$ is a Cheeger deformation by $(G, g_\textnormal{bi})$ that satisfies what we call the generalized Petersen-Wilhelm hypothesis (Definition \ref{pwhypothesisgen}), then for all $p\in M$, the orbit $G(p)$ is normal homogeneous and $g_\ell|_p$ is canonical variation with respect to the Riemannian submersion $\pi:(M, g_M)\stackrel{\ref{RQT}}{\longrightarrow}(M/G, \overline{g})$. Moreover, if $G(p)$ is totally geodesic for all $p\in M$, then $g_\ell$ is canonical variation (Theorem \ref{cheegtocangen}). \vskip 1em

\indent We then develop a technique for finding an optimal lower bound on $\textnormal{Ric}_k$ for any Riemannian manifold $(M, g_M)$ with dimension $n\geq 4$. Specifically, for any $p\in M$, unit vector $x\in T_pM$, and $k\in\mathbb{N}$ such that $2 \leq k \leq n-2$, we prove that $\min \textnormal{Ric}_{k}(x; \bullet)= \textnormal{Ric}(x)-\max \textnormal{Ric}_{n-1-k}(x; \bullet)$ (Theorem \ref{Ricklb}). \vskip 1em

\indent From there, letting $\mathbb{Z}_2$ act on $S^3\times S^3$ in two ways---as the antipodal map $a:(N_1, N_2)\mapsto (-N_1, -N_2)$ and as $f:(N_1, N_2)\mapsto (-N_1, N_2)$---we study for each value of $t\in (0,1)$ the manifold $\frac{S^3\times S^3}{\mathbb{Z}_2}$ paired with the unique metric $\overline{g_t}$ that makes the quotient map $ (S^3\times S^3, g_t)\longrightarrow \left(\frac{S^3\times S^3}{\mathbb{Z}_2}, \overline{g_t}\right)$ a local isometry. In particular, we establish $t$-independent and $t$-dependent upper bounds on the product $\min\textnormal{Ric}_k\left(\frac{S^3\times S^3}{\mathbb{Z}_2}, \overline{g_t}\right) \cdot \left(\textnormal{sys}_1\left(\frac{S^3\times S^3}{\mathbb{Z}_2}, \overline{g_t}\right)\right)^2$ when $k=2, 3, 4$ (Theorems \ref{mainresultspecific} and \ref{mainresult}). \vskip 1em

\indent Finally, we notice that our upper bounds are smaller than Wilhelm's upper bounds. We conclude that when restricted to the family $\left\{\left(\frac{S^3\times S^3}{\mathbb{Z}_2}, \overline{g_t}\right) \ | \ 0