Differential geometry is a branch of mathematics that studies the properties and structures of curves and surfaces using the techniques of calculus. It combines concepts from geometry and analysis to explore how shapes can bend and twist in space. This field is essential for understanding the geometry of objects in higher dimensions and has applications in various areas, including physics and engineering. One of the key concepts in differential geometry is the idea of a manifold, which is a space that locally resembles Euclidean space but can have a more complex global structure. Differential geometry also plays a crucial role in the theory of general relativity, where it helps describe the curvature of spacetime caused by mass and energy.