arXiv subjects
Publications and source records attributed to Krishnan Shankar.
We introduce a new probabilistic public-key cryptosystem which combines the main ingredients of the well-known RSA and Rabin cryptosystems. We investigate the security and performance of our new scheme in comparison to the other two.
In a recent article, the authors constructed a six-parameter family of highly connected 7-manifolds which admit an SO(3)-invariant metric of non-negative sectional curvature. Each member of this family is the total space of a Seifert fibration with generic fibre $S^3$ and, in particular, has the cohomology ring of an $S^3$-bundle over $S^4$. In the present article, the linking form of these manifolds is computed and used to demonstrate that the family contains infinitely many manifolds which are not even homotopy equivalent to an $S^3$-bundle over $S^4$, the first time that any such spaces have been shown to admit non-negative sectional curvature.
In this article, a six-parameter family of highly connected 7-manifolds which admit an SO(3)-invariant metric of non-negative sectional curvature is constructed and the Eells-Kuiper invariant of each is computed. In particular, it follows that all exotic spheres in dimension 7 admit an SO(3)-invariant metric of non-negative curvature.
The reach of a submanifold is a crucial regularity parameter for manifold learning and geometric inference from point clouds. This paper relates the reach of a submanifold to its convexity defect function. Using the stability properties of convexity defect functions, along with some new bounds and the recent submanifold estimator of Aamari and Levrard [Ann. Statist. 47 177-204 (2019)], an estimator for the reach is given. A uniform expected loss bound over a C^k model is found. Lower bounds for the minimax rate for estimating the reach over these models are also provided. The estimator almost achieves these rates in the C^3 and C^4 cases, with a gap given by a logarithmic factor.
In this paper, we study smooth, semi-free actions on closed, smooth, simply connected manifolds, such that the orbit space is a smoothable manifold. We show that the only simply connected $5$-manifolds admitting a smooth, semi-free circle action with fixed-point components of codimension $4$ are connected sums of $S^3$-bundles over $S^2$. Furthermore, the Betti numbers of the $5$-manifolds and of the quotient $4$-manifolds are related by a simple formula involving the number of fixed-point components. We also investigate semi-free $S^3$ actions on simply connected $8$-manifolds with quotient a $5$-manifold and show, in particular, that there are strong restrictions on the topology of the $8$-manifold.
Let $M$ be a complete Riemannian manifold and suppose $p\in M$. For each unit vector $v \in T_p M$, the $\textit{Jacobi operator}$, $\mathcal{J}_v: v^\perp \rightarrow v^\perp$ is the symmetric endomorphism, $\mathcal{J}_v(w) = R(w,v)v$. Then $p$ is an $\textit{isotropic point}$ if there exists a constant $κ_p \in \mathbf{R}$ such that $\mathcal{J}_v = κ_p \textit{ Id}_{v^\perp}$ for each unit vector $v \in T_pM$. If all points are isotropic, then $M$ is said to be isotropic; it is a classical result of Schur that isotropic manifolds of dimension at least 3 have constant sectional curvatures. In this paper we consider $\textit{almost isotropic manifolds}$, i.e. manifolds having the property that for each $p \in M$, there exists a constant $κ_p \in \mathbb{R}$, such that the Jacobi operators $\mathcal{J}_v$ satisfy $\text{rank}(\mathcal{J}_v - κ_p \textit{Id}_{v^\perp}) \leq 1$ for each unit vector $v \in T_pM$. Our main theorem classifies the almost isotropic simply connected Kähler manifolds, proving that those of dimension $d=2n \geq 4$ are either isometric to complex projective space or complex hyperbolic space or are totally geodesically foliated by leaves isometric to $\mathbf{C}^{n-1}.$
Rigidity results are obtained for Riemannian $d$-manifolds with $\sec \geqslant 1$ and spherical rank at least $d-2>0$. Conjecturally, all such manifolds are locally isometric to a round sphere or complex projective space with the (symmetric) Fubini--Study metric. This conjecture is verified in all odd dimensions, for metrics on $d$-spheres when $d \neq 6$, for Riemannian manifolds satisfying the Rakić duality principle, and for Kählerian manifolds.
We introduce and study a combinatorial optimization problem motivated by the question in the title. In the simple case where you use all objects in your room equally often, we investigate asymptotics of the optimal time to clean up in terms of the number of objects in your room. In particular, we prove a logarithmic upper bound, solve an approximate version of this problem, and conjecture a precise logarithmic asymptotic.
Magnetostrictive wires of diameter in the nanometer scale have been proposed for application as acoustic sensors [Downey et al., 2008], [Yang et al., 2006]. The sensing mechanism is expected to operate in the bending regime. In this work we derive a variational theory for the bending of magnetostrictive nanowires starting from a full 3-dimensional continuum theory of magnetostriction. We recover a theory which looks like a typical Euler-Bernoulli bending model but includes an extra term contributed by the magnetic part of the energy. The solution of this variational theory for an important, newly developed magnetostricitve alloy called Galfenol (cf. [Clark et al., 2000]) is compared with the result of experiments on actual nanowires (cf. [Downey, 2008]) which shows agreement.
In this paper we construct infinitely many examples of a Riemannian submersion from a simple, compact Lie group $G$ with bi-invariant metric onto a smooth manifold that cannot be a quotient of $G$ by a group action. This partially addresses a question of K. Grove's about Riemannian submersions from Lie groups.
In this paper we extend the concept of a conjugate point in a Riemannian manifold to complete length spaces (also known as geodesic spaces). In particular, we introduce symmetric conjugate points and ultimate conjugate points. We then generalize the long homotopy lemma of Klingenberg to this setting as well as the injectivity radius estimate also due to Klingenberg which was used to produce closed geodesics or conjugate points on Riemannian manifolds. Our versions apply in this more general setting. We next focus on ${\rm CBA}(κ)$ spaces, proving Rauch-type comparison theorems. In particular, much like the Riemannian setting, we prove an Alexander-Bishop theorem stating that there are no ultimate conjugate points less than $π$ apart in a ${\rm CBA}(1)$ space. We also prove a relative Rauch comparison theorem to precisely estimate the distance between nearby geodesics. We close with applications and open problems.
The k-dimensional Dehn (or isoperimetric) function of a group bounds the volume of efficient ball-fillings of k-spheres mapped into k-connected spaces on which the group acts properly and cocompactly; the bound is given as a function of the volume of the sphere. We advance significantly the observed range of behavior for such functions. First, to each non-negative integer matrix P and positive rational number r, we associate a finite, aspherical 2-complex X_{r,P} and calculate the Dehn function of its fundamental group G_{r,P} in terms of r and the Perron-Frobenius eigenvalue of P. The range of functions obtained includes x^s, where s is an arbitrary rational number greater than or equal to 2. By repeatedly forming multiple HNN extensions of the groups G_{r,P} we exhibit a similar range of behavior among higher-dimensional Dehn functions, proving in particular that for each positive integer k and rational s greater than or equal to (k+1)/k there exists a group with k-dimensional Dehn function x^s. Similar isoperimetric inequalities are obtained for arbitrary manifold pairs (M,\partial M) in addition to (B^{k+1},S^k).
The known manifolds of positive sectional curvature are either homogeneous spaces or biquotients, i.e. quotients of a compact Lie group by a group acting on the left and right simultaneously. The full isometry group of the homogeneous metrics of positive curvature were determined by K.Shankar. Here we determine the isometry group of some of the biquotients due to Eschenburg and Bazaikin. As an application we obtain, as in the homogeneous case, more counterexamples to the Chern conjecture, which states that an abelian subgroup of the fundamental group of a positively curved manifold is cyclic.
Let M be a complete Riemannian manifold whose sectional curvature is bounded above by 1. We say that M has positive spherical rank if along every geodesic one hits a conjugate point at t=π. The following theorem is then proved: If M is a complete, simply connected Riemannian manifold with upper curvature bound 1 and positive spherical rank, then M is isometric to a compact, rank one symmetric space (CROSS) i.e., isometric to a sphere, complex projective space, quaternionic projective space or to the Cayley plane. The notion of spherical rank is analogous to the notions of Euclidean rank and hyperbolic rank studied by several people (see references). The main theorem is proved in two steps: first we show that M is a so called Blaschke manifold with extremal injectivity radius (equal to diameter). Then we prove that such M is isometric to a CROSS.
We compute the Eells-Kuiper invariant of the Berger manifold SO(5)/SO(3) and determine that it is diffeomorphic to the total space of an S^3-bundle over S^4. This answers a question raised by K. Grove and W. Ziller.
There has been renewed interest in $\mathbb{S}^3$-bundles over $\mathbb{S}^4$ since K. Grove and W. Ziller constructed metrics on nonnegative curvature on the total spaces of these bundles. In this paper we write down necessary and sufficient conditions for a CW complex to be homotopy equivalent to such a bundle. We also show that for a manifold homotopy equivalent to such a bundle, in certain cases, there is no obstruction to homeomorphism. We use this to show that the Berger manifold, $\text{Sp}(2)/\text{Sp}(1)$, is PL-homeomorphic to such a bundle. This question was raised in the paper by Grove and Ziller since this manifold, which admits a normal homogeneous metric of positive sectional curvature, has the cohomology ring of such a bundle. The principal technique used is the study of the Serre spectral sequence of various fibrations.