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Register a new userIsometric immersions with flat normal bundle between space forms
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We investigate the behavior of the second fundamental form of an isometric immersion of a space form with negative curvature into a space form so that the extrinsic curvature is negative. If the immersion has flat normal bundle, we prove that its second fundamental form grows exponentially.
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Kirigami is the art of cutting paper to make it articulated and deployable, allowing for it to be shaped into complex two and three-dimensional geometries. The mechanical response of a kirigami sheet when it is pulled at its ends is enabled and limited by the presence of cuts that serve to guide the possible non-planar deformations. Inspired by the geometry of this art form, we ask two questions: (i) What is the shortest path between points at which forces are applied? (ii) What is the nature of the ultimate shape of the sheet when it is strongly stretched? Mathematically, these questions are related to the nature and form of geodesics in the Euclidean plane with linear obstructions (cuts), and the nature and form of isometric immersions of the sheet with cuts when it can be folded on itself. We provide a constructive proof that the geodesic connecting any two points in the plane is piecewise polygonal. We then prove that the family of polygonal geodesics can be simultaneously rectified into a straight line by flat-folding the sheet so that its configuration is a (non-unique) piecewise affine planar isometric immersion.
We revisit the classical problem due to Weyl, as well as its generalisations, concerning the isometric immersions of $mathbb{S}^2$ into simply-connected $3$-dimensional Riemannian manifolds with non-negative Gauss curvature. A sufficient condition is exhibited for the existence of global $C^{1,1}$-isometric immersions. Our developments are based on the framework `{a} la Labourie (Immersions isom{e}triques elliptiques et courbes pseudo-holomorphes, J. Diff. Geom. 30 (1989), 395--424) of studying isometric immersions using $J$-holomorphic curves. We obtain along the way a generalisation of a classical theorem due to Heinz and Pogorelov.
The aim of this paper is to study Sasakian immersions of (non-compact) complete regular Sasakian manifolds into the Heisenberg group and into $ mathbb{B}^Ntimes mathbb{R}$ equipped with their standard Sasakian structures. We obtain a complete classification of such manifolds in the $eta$-Einstein case.
Let $Kbackslash G$ be an irreducible Hermitian symmetric space of noncompact type and $Gamma ,subset, G$ a closed torsionfree discrete subgroup. Let $X$ be a compact Kahler manifold and $rho, :, pi_1(X, x_0),longrightarrow, Gamma$ a homomorphism such that the adjoint action of $rho(pi_1(X, x_0))$ on $text{Lie}(G)$ is completely reducible. A theorem of Corlette associates to $rho$ a harmonic map $X, longrightarrow, Kbackslash G/Gamma$. We give a criterion for this harmonic map to be holomorphic. We also give a criterion for it to be anti--holomorphic.
We explore the relation among volume, curvature and properness of a $m$-dimensional isometric immersion in a Riemannian manifold. We show that, when the $L^p$-norm of the mean curvature vector is bounded for some $m leq pleq infty$, and the ambient manifold is a Riemannian manifold with bounded geometry, properness is equivalent to the finiteness of the volume of extrinsic balls. We also relate the total absolute curvature of a surface isometrically immersed in a Riemannian manifold with its properness. Finally, we relate the curvature and the topology of a complete and non-compact $2$-Riemannian manifold $M$ with non-positive Gaussian curvature and finite topology, using the study of the focal points of the transverse Jacobi fields to a geodesic ray in $M$ . In particular, we have explored the relation between the minimal focal distance of a geodesic ray and the total curvature of an end containing that geodesic ray.
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