Narrow Results

Consider a Riemannian manifold N equipped with an additional geometric structure, such as a Kähler structure or a quaternionic Kähler structure, and a hypersurface M in N. The geometric structure induces a decomposition of the tangent bundle TM of M into subbundles. A natural problem is to classify all hypersurfaces in N for which the second fundamental form of M preserves these subbundles. This problem is reasonably well understood for Riemannian symmetric spaces of rank one, but not for higher rank symmetric spaces. A general treatment of this problem for higher rank symmetric spaces is out of reach at present, and therefore it is desirable to understand this problem better in a few special cases. Due to some conceptual differences between symmetric spaces of compact type and of noncompact type it appears that one needs to consider these two cases separately. In this paper we investigate this problem for the rank two symmetric space SU_{2, m}/S(U₂U_m) of noncompact type.

In the study of discontinuous groups for non-Riemannian homogeneous spaces, the idea of “continuous analogue” gives a powerful method (T. Kobayashi [Math. Ann. 1989]). For example, a semisimple symmetric space $G/H$ admits a discontinuous group which is not virtually abelian if and only if $G/H$ admits a proper $SL(2,ℝ)$-action (T. Okuda [J. Differ. Geom. 2013]). However, the action of discrete subgroups is not always approximated by that of connected groups. In this paper, we show that the theorem cannot be extended to general homogeneous spaces $G/H$ of reductive type. We give a counterexample in the case $G=SL(5,ℝ)$.

Starting with the complexes, quaternions and the octonions we study five families of groups and homogeneous spaces each building geometrically on the previous families. For each family a particular theorem is studied from the perspective of the structure developed. The evolution of the spaces and the corresponding results form a sort of mathematical memoir of a career spanning more than 50 years.

We discuss the target space pseudoduality in supersymmetric sigma models on symmetric spaces using two different methods, orthonormal coframe and component expansion. These two methods yield similar results to the classical cases with the exception that commuting bracket relations in classical case turn out to be anticommuting ones because of the appearance of Grassmann numbers. In component expansion method it is understood that constraint relations in case of non-mixing pseudoduality are the remnants of mixing pseudoduality. Once mixing terms are included in the pseudoduality relations, the constraint relations disappear.

We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a tangent space. Among the Lie algebra weight systems, they are exactly characterized by having the symmetries of the Riemann curvature tensor.

Let $(X,{𝒯})$ be a topological space. By the Skula topology (or the $b$-topology) on $X$, we mean the topology $b({𝒯})$ on $X$ with basis the collection of all ${𝒯}$-locally closed sets of $X$, the resulting space $(X,b({𝒯}))$ will be denoted by $b(X)$. We show that the following results hold:

• (1)$b(X)$ is an Alexandroff space if and only if the $T_0$-reflection ${\mathrm{\text{T}}}_0(X)$ of $X$ is a $T_D$-space.
• (2)$b(X)$ is a Noetherian space if and only if ${\mathrm{\text{T}}}_0(X)$ is finite.
• (3)If we denote by $X^{*}$ the Alexandroff extension of $X$, then $b(X^{*})={(b(X))}^{*}$ if and only if $(X,{𝒯})$ is a Noetherian quasisober space.

We also give an alternative proof of a result due to Simmons concerning the iterated Skula spaces, namely, $b(b(b(X)))=b(b(X))$.

A space is said to be clopen if its open sets are also closed. In [R. E. Hoffmann, Irreducible filters and sober spaces, Manuscripta Math. 22 (1977) 365–380], Hoffmann introduced a refinement clopen topology $\mathrm{\text{Clop}}({𝒯})$ of ${𝒯}$: The indiscrete components of $\mathrm{\text{Clop}}(X)$ are of the form $C_x=\overline{\{x\}}\cap {𝒪}(x)$, where $x\in X$ and ${𝒪}(x)$ is the intersection of all open sets of $X$ containing $x$ (equivalently, $C_x=\{y\in X:\overline{\{x\}}=\overline{\{y\}}\}$). We show that $\mathrm{\text{Clop}}(X)=b(b(X))$

Windowing a Fourier transform is a useful tool, which gives us the similarity between the signal and time frequency signal, and it allows to get sense when/where certain frequencies occur in the input signal, this method was introduced by Dennis Gabor. In this paper, we generalize the classical Gabor–Fourier transform (GFT) to the Riemannian symmetric space calling it the Helgason–Gabor–Fourier transform (HGFT). We prove several important properties of HGFT like the reconstruction formula, the Plancherel formula and Parseval formula. Finally, we establish some local uncertainty principle such as Benedicks-type uncertainty principle.

In this paper, we will show how particular formation of G-invariant metrics on irreducible symmetric spaces leads us to have a different approach to Ricci flow as a weakly parabolic system which happens to become a ODE under these spaces.

This note investigates the multiplicity problem of principal curvatures of equifocal hypersurfaces in simply connected rank 1 symmetric spaces. Using Clifford representation theory, and the author also constructs infinitely many equifocal hypersurfaces in the symmetric spaces.

Let $X=G/K$ be a higher rank symmetric space of non-compact type where $G={\mathrm{\text{Isom}}}^0(X)$. We define the splitting rank of $X$, denoted by $\mathrm{\text{srk}}(X)$, to be the maximal dimension of a totally geodesic submanifold $Y\subset X$ which splits off an isometric $ℝ$-factor. We compute explicitly the splitting rank for each irreducible symmetric space. For an arbitrary (not necessarily irreducible) symmetric space, we show that the comparison map $\eta :H_{c,b}^{\ast }(G,ℝ)\to H_c^{\ast }(G,ℝ)$ is surjective in degrees $\ast \ge \mathrm{\text{srk}}(X)+2$, provided $X$ has no direct factors of ${ℍ}^2$, $\mathrm{\text{SL}}(3,ℝ)/\mathrm{\text{SO}}(3)$, $\mathrm{\text{Sp}}(2,ℝ)/U(2)$, $G_2^2/\mathrm{\text{SO}}(4)$ and $\mathrm{\text{SL}}(4,ℝ)/\mathrm{\text{SO}}(4)$. This generalizes the result of [J.-F. Lafont and S. Wang, Barycentric straightening and bounded cohomology, to appear in J. Eur. Math. Soc.] regarding Dupont’s problem.

The motivation of this paper is to introduce Henstock–Orlicz space with non-absolute integrable functions. We prove that $C_0^{\infty }$ is dense in the Henstock–Orlicz space, which is not dense in the classical Orlicz space.

For k ≥ 2, we exhibit complete k-curvature homogeneous neutral signature pseudo-Riemannian manifolds which are not locally affine homogeneous (and hence not locally homogeneous). The curvature tensor of these manifolds is modeled on that of an indecomposible symmetric space. All the local scalar Weyl curvature invariants of these manifolds vanish.

We get several identities of differential operators in determinantal form. These identities are non-commutative versions of the formula of Cauchy-Binet or Laplace expansions of determinants, and if we take principal symbols, they are reduced to such classical formulas. These identities are naturally arising from the generators of the rings of invariant differential operators over symmetric spaces, and have strong resemblance to the classical Capelli identities. Thus we call those identities the Capelli identities for symmetric pairs.

The study of positive sectional curvature is one of the oldest pursuits in Riemannian geometry, but despite the considerable efforts of many researchers, basic questions remain unanswered. In this lecture we will briefly summarize the state of knowledge in this area and outline the techniques which have had success. These techniques include geodesic and comparison methods, minimal surface methods, and Ricci flow. We will then describe our recent work (see [18], [21], [22]) which uses the Ricci flow to resolve the differentiable sphere theorem; that is, the complete classification of manifolds whose sectional curvatures are 1/4-pinched.