In this paper is studied the configuration of lines of curvature near a Whitney umbrella which is the unique stable singularity for maps of surfaces into $R^3$. The pattern of such configuration is established and characterized in terms of the 3-jet of the map. The result is used to establish an expression for the Euler-Poincaré characteristic in terms of the number of umbilics and umbrellas.

We give a transparent description of the one-fold smooth suspension of Fintushel-Stern's exotic involution on the 4-sphere. Moreover we prove that any two involutions of the 4-sphere are stably (i.e., after one-fold suspension) smoothly conjugated if and only if the corresponding quotient spaces (real homotopy projective spaces) are stably diffeomorphic. We use the Atiyah-Patodi-Singer eta-invariant to detect smooth structures on homotopy projective spaces and prove that any homotopy projective space is detected in this way in dimensions 5 and 6.

Two basic Lie-invariant forms uniquely defining a generic (hyper)surface in Lie sphere geometry are introduced. Particularly interesting classes of surfaces associated with these invariants are considered. These are the diagonally cyclidic surfaces and the Lie-minimal surfaces, the latter being the extremals of the simplest Lie-invariant functional generalizing the Willmore functional in conformal geometry. Equations of motion of a special Lie sphere frame are derived, providing a convenient unified treatment of surfaces in Lie sphere geometry. In particular, for diagonally cyclidic surfaces this approach immediately implies the stationary modified Veselov-Novikov equation, while the case of Lie-minimal surfaces reduces in a certain limit to the integrable coupled Tzitzeica system. In the framework of the canonical correspondence between Hamiltonian systms of hydrodynamic type and hypersurfaces in Lie sphere geometry, it is pointed out that invariants of Lie-geometric hypersurfaces coincide with the reciprocal invariants of hydrodynamic type systems. Integrable evolutions of surfaces in Lie sphere geometry are introduced. This provides an interpretation of the simplest Lie-invariant functional as the first local conservation law of the (2+1)-dimensional modified Veselov-Novikov hierarchy. Parallels between Lie sphere geometry and projective differential geometry of surfaces are drawn in the conclusion.

The main aim of this paper is to compute the index and the nullity of the identity map of $S^n\times S^m$ and $S^n\times T^m$. In order to obtain this we establish a rather general result on the spectrum of the Hodge-Laplacian on $k$-forms on a product manifold, which could prove useful in other contexts.

Suppose the fractional integration operator $I^{\sigma}$ is generated by the sequence $\{(k+1)^{-\sigma}\}$ in the setting of Laguerre and Hermite expansions. Then, via projection formulas, the problem of the norm boundedness of $I^{\sigma}$ is reduced to the well-known fractional integration on the half-line. A corresponding result with respect to the modified Hankel transform is derived and its connection with the Laguerre fractional integration is indicated.

There exists a Riemannian metric on the real projective space such that the first eigenvalue coincides with that of its Riemannian universal cover, if the dimension is bigger than 2. For the proof, we deform the canonical metric on the real projective space. A similar result is obtained for lens spaces, as well as for closed Riemannian manifolds with Riemannian double covers. As a result, on a non-orientable closed manifold other than the real projective plane, there exists a Riemannian metric such that the first eigenvalue coincides with that of its Riemannian double cover.

By using moving frames and directred digraphs, we study invariant (1,2)-symplectic structures on complex flag manifolds. Let $F$ be a flag manifold with height $k-1$. We show that there is a $k$-dimensional family of invariant (1,2)-symplectic metrics of any parabolic structure on $F$. We also prove any invariant almost complex structure $J$ on $F$ with height 4 admits an invariant (1,2)-symplectic metric if and only if $J$ is parabolic or integrable.

For an isometric immersion $f$ of a flat torus into the unit 3-sphere, we show that if the mean curvature of $f$ is not constant, then the immersion $f$ admits a nontrivial isometric deformation preserving the total mean curvature.

It was established by X. Mo and the author that the dimension of each irreducible component of the moduli space $\mathcal{M}_{d,g}(X)$ of branched superminimal immersions of degree $d$ from a Riemann surface $X$ of genus $g$ into $C P^3$ lay between $2d-4g+4$ and $2d-g+4$ for $d$ sufficiently large, where the upper bound was always assumed by the irreducible component of {\it totally geodesic} branched superminimal immersions and the lower bound was assumed by all {\it nontotally geodesic} irreducible components of $\mathcal{M}_{6,1}(T)$ for any torus $T$. It is shown, via deformation theory, in this note that for $d=8g+1+3k$, $k\geq 0$, and any Riemann surface $X$ of $g\geq 1$, the above lower bound is assumed by at least one irreducible component of $\mathcal{M}_{d,g}(X)$.

The disjoint union of mapping class groups forms a braided monoidal category. We give an explicit expression of braidings in terms of both their actions on the fundamental group of the surface and the standard Dehn twists. This braided monoidal category gives rise to a double loop space. We prove that the action of little 2-cube operad does not extend to the action of little 3-cube operad by showing that the Browder operation induced by 2-cube operad action is nontrivial. A rather simple expression of Reshetikhin-Turaev representation is given for the sixteenth root of unity in terms of matrices with entries of complex numbers. We show by matrix calculation that this representation is symmetric with respect to the braid structure.