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## V. CONCLUSIONS 

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In ( n + 1)-dimensions, when the ( n − 1)-sphere of black hole event horizons is replaced by an ( n − 1)-dimensional hypersurface with zero or negative constant curvature, the black hole is referred to as a topological black brane. The construction and analysis of these exotic black branes in AdS space is a subject of much recent interest. This is primarily due to their relevance for the AdS/CFT correspondence. In this paper, we further generalized these exotic black brane solutions by including a dilaton and the electrodynamic fields in the action. We obtained a new class of ( n + 1)-dimensional ( n ≥ 3) topological black brane solutions in Einstein-Maxwell-dilaton gravity in the background of AdS spaces. Indeed, the dilaton potential plays a crucial role in the existence of these black brane solutions, as the negative cosmological constant does in the Einstein-Maxwell theory. In the absence of a dilaton field ( α = γ = 0), our solutions reduce to the ( n + 1)-dimensional topological black brane solutions presented in [14]. We computed the entropy, temperature, charge, mass, and electric potential of the topological dilaton black branes and found that these quantities satisfy the first law of thermodynamics. 

For the thermodynamical analysis of the black branes, we divide the black branes into three classes. The black brane is said to be ”small” when the radius of event horizon is much smaller than the parameter b , it is called ”medium” if r$_{+}$ and b have the same order and it is called ”large” for large values of r$_{+}$ with respect to b . The radius of event horizon is always larger or equal to r$_{ext}$ for the black branes with flat horizon, while it is in the range 

