In mathematics, an analytic manifold, also known as a 
  
    
      
        
          C
          
            ω
          
        
      
    
    {\displaystyle C^{\omega }}
  
 manifold, is a differentiable manifold with analytic transition maps. The term usually refers to real analytic manifolds, although complex manifolds are also analytic. In algebraic geometry, analytic spaces are a generalization of analytic manifolds such that singularities are permitted.
 For 
  
    
      
        U
        ⊆
        
          
            R
          
          
            n
          
        
      
    
    {\displaystyle U\subseteq \mathbb {R} ^{n}}
  
, the space of analytic functions, 
  
    
      
        
          C
          
            ω
          
        
        (
        U
        )
      
    
    {\displaystyle C^{\omega }(U)}
  
, consists of infinitely differentiable functions 
  
    
      
        f
        :
        U
        →
        
          R
        
      
    
    {\displaystyle f:U\to \mathbb {R} }
  
, such that the Taylor series
 
  
    
      
        
          T
          
            f
          
        
        (
        
          x
        
        )
        =
        
          ∑
          
            
              |
            
            α
            
              |
            
            ≥
            0
          
        
        
          
            
              
                D
                
                  α
                
              
              f
              (
              
                
                  x
                  
                    0
                  
                
              
              )
            
            
              α
              !
            
          
        
        (
        
          x
        
        −
        
          
            x
            
              0
            
          
        
        
          )
          
            α
          
        
      
    
    {\displaystyle T_{f}(\mathbf {x} )=\sum _{|\alpha |\geq 0}{\frac {D^{\alpha }f(\mathbf {x_{0}} )}{\alpha !}}(\mathbf {x} -\mathbf {x_{0}} )^{\alpha }}
  
 
 converges to 
  
    
      
        f
        (
        
          x
        
        )
      
    
    {\displaystyle f(\mathbf {x} )}
  
 in a neighborhood of 
  
    
      
        
          
            x
            
              0
            
          
        
      
    
    {\displaystyle \mathbf {x_{0}} }
  
, for all 
  
    
      
        
          
            x
            
              0
            
          
        
        ∈
        U
      
    
    {\displaystyle \mathbf {x_{0}} \in U}
  
. The requirement that the transition maps be analytic is significantly more restrictive than that they be infinitely differentiable; the analytic manifolds are a proper subset of the smooth, i.e. 
  
    
      
        
          C
          
            ∞
          
        
      
    
    {\displaystyle C^{\infty }}
  
, manifolds. There are many similarities between the theory of analytic and smooth manifolds, but a critical difference is that analytic manifolds do not admit analytic partitions of unity, whereas smooth partitions of unity are an essential tool in the study of smooth manifolds. A fuller description of the definitions and general theory can be found at differentiable manifolds, for the real case, and at complex manifolds, for the complex case.
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