The work is devoted to the search for explicit isometric embeddings of a metric corresponding to the collapse of spherically symmetric matter with the formation of a black hole. Two approaches are considered: in the first, the embedding is constructed for the whole manifold at once; in the second, the idea of a junction of solutions, obtained separately for areas inside and outside the dust ball, is used. In the framework of the first approach, a global smooth embedding in 7D space with a signature (2  +  5) was constructed. It corresponds to the formation of the horizon as a result of matter falling from infinity. The second approach generally leads to an embedding in 7D space with the signature (1  +  6). This embedding corresponds to the case when matter flies out of a white hole with the disappearance of its horizon, after which the radius of the dust ball reaches its maximum, and then a collapse occurs with the formation of the horizon of a black hole. The embedding obtained is not smooth everywhere—it contains a kink on the edge of the dust ball, and also, it is not quite global. In the particular case, when the maximum radius of the dust ball coincides with the radius of the horizon, it is possible to construct a global smooth embedding in a flat 6D space with a signature (1  +  5).

We propose an analytical method to describe a matter density profile near a galaxy center. The description is based on the study of the distribution function of particles over possible trajectories. We establish a relation between the central slope of density profile and the near-origin behavior of the angular momentum distribution function. We consider both a spherically symmetric (on average) matter distribution as well as deviations from it. If the density profile forms in a background of spherical gravitation potential then a core-type distribution arises. A regular matter may behave in such way if the background potential was formed by the dark matter. In the presence of deviation from spherical symmetry the formation of cusp-type distribution is possible. Moreover, a reduction of spherical symmetry to the axial one leads to a less steep cusp profile. The complete symmetry breaking (which corresponds, in particular, to the common setup of numerical simulations), leads to a steeper cusp profile.

The work is devoted to the construction of explicit embeddings for the metrics of the black holes, formed by nonsingular matter distribution. One of the possible examples of such type of solutions is regular black holes. Using the existing classification of minimal symmetric embeddings of the Schwarzschild metric as a base, we construct embeddings for regular black holes with de Sitter interior. Another simple example is black hole, formed by collapsing homogeneous spherically symmetric cloud of dustlike matter. We discuss embeddings for two variants of such black holes -the one with the eternally existing horizon, when dust ball never leaves the interior of the horizon, and another variant with the dynamically forming horizon.

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