The de Sitter space is defined as a solution to Einstein equations with positive cosmological constant. In this thesis we study (2+1)-dimensional de Sitter manifolds whit (-++) signature embedded in a flat space-time with (-+++) signature. As the first step to realize symmetry group of the manifold, one-parameter subgroups of SO(3,1) symmetry group has been classified. These subgroups divide into two distinct types: permissible and impermissible. The permissible types result in the metric tensor of embedding space in such a way that its determinate is compatible with the (-+++) signature of metric. On the other hand, in impermissible type the determinate of metric tensor is positive which is obviously incompatible with the signature of embedding space. By realizing all permissible types of the one-parameter subgroups for SO(3,1), one can determine the isometries of the manifold. The generator of infinitesimal displacement along such directions is called a Killing vector. One can compactify the space along one of these isometries. The norm of the Killing vector and other properties of embedding space determine whether it is possible to have a black hole solution or not. The norm of both of Killing vectors in de Sitter space show that it is impossible to find a black hole solution by this way. Whereas, in anti-de Sitter space, which is defined by negative cosmological constant, this process results in black hole solution. One can show that this different behavior is an inheritance of different embedding spaces. The de Sitter manifold can be defined by different coordinates systems leading to metrics with different properties. Beside the familiar metrics, we introduce another one which is completely covered by “global metric” in (2+1) dimensions. It means that the domain of validity of this metric is a subset of the global one. We called the common region which is covered by both metrics as “intersection region” which is restricted by null geodesies. It is not straightforward however to identify a coordinate transformation between these two metrics in this region. In addition, these two metrics show completely different properties. As an example, the global metric is time-dependent because the spherical part of this metric expands monotonically in time. But the other metric is periodic in time and its volume vanishes periodically in spatial distances. Also observers, corresponding to these metrics, make different judgments about longevity of the intersection region. A local observer using the second metric can “see” the whole intersection region which exists forever. But from the point of view of the “global” observer, the intersection region disappears gradually by the rate which depends on the position of the observer. The causal structures and all these differences can be shown in a Penrose diagrams. It is specially an important example in de Sitter space because brings up the following question “in the absence of coordinate transformation between two distinct metrics how could the observers achieve a common perception about their surrounding events?”. As a practical example, how the observer in one metric could determine his/her wave function by using a given wave function for another metric. Keywords: Anti De Sitter Space,De Sitter Space and Different Metrics,One-parameter Subgroup of SO(2,2),SO(3,1) and SO(4),Isometries,Identification,BTZ Black Hole,Intersection Region,Thermodynamics of Black Holes,Black Hole and Cosmological Event Horizon