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We construct two new versions of the c-map which allow us to obtain the target manifolds of hypermultiplets in Euclidean theories with rigid N = 2 supersymmetry. While the Minkowskian para-c-map is obtained by dimensional reduction of the Minkowskian vector multiplet lagrangian over time, the Euclidean para-c-map corresponds to the dimensional reduction of the Euclidean vector multiplet lagrangian. In both cases the resulting hypermultiplet target spaces are para-hyper-Kähler manifolds. We review and prove the relevant results of para-complex and para-hypercomplex geometry. In particular, we give a second, purely geometrical construction of both c-maps, by proving that the cotangent bundle N = T * M of any affine special (para-)Kähler manifold M is parahyper-Kähler.

[1] In Global Navigation Satellite Systems (GNSS) using L band frequencies, the ionosphere causes signal delays that correspond to link-related range errors of up to 100 m. Whereas this error can be corrected in dual-frequency measurements by a linear combination of L1 and L2 phases, in single-frequency measurements, additional information is needed to mitigate the ionospheric error which is proportional to the total electron content (TEC) of the ionosphere. This information can be provided by TEC maps deduced from corresponding GNSS measurements or from model values. Besides direct range error correction in navigation and remote sensing applications, TEC or electron density models play a key role in ionospheric monitoring and forecasting. In this paper we discuss the development and use of TEC models for calibrating TEC, reconstructing reliable TEC maps, and forecasting TEC behavior based on GNSS measurements. European and global TEC maps and corresponding 1 h ahead forecasts are distributed via the operational space weather and ionosphere data service (http://swaciweb.dlr.de) to the international community. The Neustrelitz TEC Model is a basic approach for a family of regional and global TEC models used in different types of applications. The model approximates typical TEC variations depending on the location, time, and level of solar activity with only a few coefficients.

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