### The classification of 4-dimensional Kähler manifolds with small eigenvalue of the Dirac operator.

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We estimate from below by geometric data the eigenvalues of the periodic Sturm-Liouville operator $-4{d}^{2}/d{s}^{2}+{\kappa}^{2}\left(s\right)$ with potential given by the curvature of a closed curve.

Any 7-dimensional cocalibrated ${G}_{2}$-manifold admits a unique connection $\nabla $ with skew symmetric torsion (see [8]). We study these manifolds under the additional condition that the $\nabla $-Ricci tensor vanish. In particular we describe their geometry in case of a maximal number of $\nabla $-parallel vector fields.

Using the Cartan method O. Boruvka (see [B1], [B2]) studied superminimal surfaces in four-dimensional space forms. In particular, he described locally the family of all superminimal surfaces and classified all of them with a constant radius of the indicatrix. We discuss the mentioned results from the point of view of the twistor theory, providing some new proofs. It turns out that the superminimal surfaces investigated by geometers at the beginning of this century as well as by O. Boruvka...

In some other context, the question was raised how many nearly Kähler structures exist on the sphere ${\mathbb{S}}^{6}$ equipped with the standard Riemannian metric. In this short note, we prove that, up to isometry, there exists only one. This is a consequence of the description of the eigenspace to the eigenvalue $\lambda =12$ of the Laplacian acting on $2$-forms. A similar result concerning nearly parallel ${\mathrm{G}}_{2}$-structures on the round sphere ${\mathbb{S}}^{7}$ holds, too. An alternative proof by Riemannian Killing spinors is also indicated.

We give an introduction into and exposition of Seiberg-Witten theory.

We prove that the ring ℝ[M] of all polynomials defined on a real algebraic variety $M\subset {\mathbb{R}}^{n}$ is dense in the Hilbert space ${L}^{2}(M,{e}^{-{\left|x\right|}^{2}}d\mu )$, where dμ denotes the volume form of M and $d\nu ={e}^{-{\left|x\right|}^{2}}d\mu $ the Gaussian measure on M.

[For the entire collection see Zbl 0699.00032.] The author considers the conformal relation between twistors and spinors on a Riemannian spin manifold of dimension $n\ge 3$. A first integral is constructed for a twistor spinor and various geometric properties of the spin manifold are deduced. The notions of a conformal deformation and a Killing spinor are considered and such a deformation of a twistor spinor into a Killing spinor and conditions for the equivalence of these quantities is indicated.

A G-structure on a Riemannian manifold is said to be integrable if it is preserved by the Levi-Civita connection. In the presented paper, the following non-integrable G-structures are studied: SO(3)-structures in dimension 5; almost complex structures in dimension 6; G${}_{2}$-structures in dimension 7; Spin(7)-structures in dimension 8; Spin(9)-structures in dimension 16 and F${}_{4}$-structures in dimension 26. G-structures admitting an affine connection with totally skew-symmetric torsion are characterized....

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