By the end of this course, students should be able to:

• Understand  Lie groups and Lie algebras and actions of Lie groups on manifolds.
• Comprehend the theory of principal fiber bundles and learn connections and curvature on principal bundles.
• Define almost complex structures and integrability conditions and study both almost complex and complex manifolds. 
• Learn Hermitian metrics and understand their compatibility with complex structures.
• Learn Kähler manifolds and analyze their properties.
• Understand the role of the Kähler form and its implications in geometry.
• Define symmetric spaces and investigate their classification.
• Analyze geometric and algebraic properties of symmetric spaces and study examples of compact and noncompact symmetric space.