In this paper, we define a new class of Riemannian submanifolds which we call arid submanifolds. A Riemannian submanifold is called an arid submanifold if no nonzero normal vectors are invariant under the full slice representation. We see that arid submanifolds are a generalization of weakly reflective submanifolds, and arid submanifolds are minimal submanifolds. We also introduce an application of arid submanifolds to the study of left-invariant metrics on Lie groups. We give a suffcient condition for a left-invariant metric on an arbitrary Lie group to be a Ricci soliton.

We consider the cosmetic surgery problem for two-bridge knots in the 3-sphere. We first verify by using previously known results that all the two-bridge knots of at most $9$ crossings admit no purely cosmetic surgery pairs except for the knot $9_{27}$. Then we show that any two-bridge knot corresponding to the continued fraction $[0, 2x, 2, -2x, 2x, 2, -2x]$ for a positive integer $x$ admits no cosmetic surgery pairs yielding homology 3-spheres, where $9_{27}$ appears when $x = 1$. Our advantage to prove this is using the $SL(2,\mathbb{C})$ Casson invariant.

In this paper we study the LCM-stability property and other related concepts, and their universality in the case of polynomial and formal power series extensions.

Firstly, we obtain conditions for stable extendibility and extendibility of complex vector bundles over the $(2n+1)$-dimensional standard lens space $L^n(p)$ mod $p$, where $p$ is a prime. Secondly, we prove that the complexification $c(\tau_n(p))$ of the tangent bundle $\tau_n(p) (=\tau(L^n(p)))$ of $L^n(p)$ is extendible to $L^{2n+1}(p)$ if $p$ is a prime, and is not stably extendible to $L^{2n+2}(p)$ if $p$ is an odd prime and $n \ge 2p-2$. Thirdly, we show, for some odd prime $p$ and positive integers $n$ and $m$ with $m > n$, that $\tau(L^n(p))$ is stably extendible to $L^m(p)$ but is not extendible to $L^m(p)$.

We study non-isotrivial families of $K3$ surfaces in positive characteristic $p$ whose geometric generic fibers satisfy $\rho \ge 21 - 2h$ and $h \ge 3$, where $\rho$ is the Picard number and $h$ is the height of the formal Brauer group. We show that, under a mild assumption on the characteristic of the base field, they have potential supersingular reduction. Our methods rely on Maulik’s results on moduli spaces of $K3$ surfaces and the construction of sections of powers of Hodge bundles due to van der Geer and Katsura. For large $p$ and each $2 \le h \le 10$, using deformation theory and Taelman’s methods, we construct non-isotrivial families of $K3$ surfaces satisfying $\rho = 22 - 2h$.

We give a small generating set for the twist subgroup of the mapping class group of a non-orientable surface by Dehn twists. The difference between the number of the generators and a lower bound of numbers of generators for the twist subgroup by Dehn twists is one. The lower bounds is obtained from an argument of Hirose [5].

We introduce a multiple conjugation biquandle, and show that it is the universal algebra for defining a semi-arc coloring invariant for handlebody-links. A multiple conjugation biquandle is a generalization of a multiple conjugation quandle. We extend the notion of $n$-parallel biquandle operations for any integer $n$, and show that any biquandle gives a multiple conjugation biquandle with them.