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1. Affine Connection

   Linked via "curved manifolds"

   An affine connection (or simply a connection{:data-entity="connection"}) is a fundamental structure in differential geometry that generalizes the concept of directional derivatives from Euclidean space to curved manifolds. It provides a consistent mathematical framework for performing covariant differentiation of tensors, thereby defining concept…

2. Covariant Derivative

   Linked via "curved manifolds"

   The covariant derivative ($\nabla{\mu}$) is a fundamental operator in differential geometry and theoretical physics, designed to generalize the ordinary partial derivative ($\partial{\mu}$) when working on curved manifolds or in coordinate systems whose basis vectors vary from point to point. Its primary function is to define a concept of differentiation that respects the underlying geometric structure, ensuring that physical l…

3. Covariant Differentiation

   Linked via "curved manifolds"

   Covariant differentiation, symbolized generally by $\nabla$, is a central concept in differential geometry and theoretical physics, extending the notion of the ordinary partial derivative to vector fields and tensor fields defined on curved manifolds or in spaces where the coordinate system is non-Cartesian and non-inertial [7]. It fundamentally …