what is the meaning of covariant derivatives?

Introduction

Covariant derivatives are a fundamental concept in differential geometry and theoretical physics, particularly in the study of curved spaces and manifolds. They extend the notion of differentiation to curved spaces, allowing for the differentiation of tensor fields in a manner that respects the geometric structure of the space.

Key Insights

Geometric and Group Theoretical Interpretation:
- Covariant derivatives can be understood as horizontal left-invariant vector fields on superspace derived from a (super)Lie group, which also functions as a principal bundle with a canonical connection.
Extension to Generalized Covariant Derivatives:
- The classical covariant derivative, traditionally acting on components, is extended to a generalized form that can act on any geometric quantity, including base vectors, vectors, and tensors. This extension simplifies tensor analysis in curved coordinate systems .
Covariant Derivatives in Curved Spaces:
- The generalized covariant derivative is further extended to curved surfaces, establishing a covariant differential transformation group with orthogonal duality. This framework simplifies tensor analysis and constructs differential and integral invariabilities on curved surfaces.
Covariant Derivatives of Eigenfunctions:
- On a complete, connected Riemannian manifold with nonzero constant sectional curvature, the covariant derivative of an eigenfunction for the Laplace-Beltrami operator along every parallel tensor is a scalar multiple of the eigenfunction. This scalar is a polynomial dependent on the eigenvalue.
General Covariant Derivatives for General Connections:
- General covariant derivatives of vertical-valued tensor fields with respect to a general connection on a fibered manifold and a classical connection on the base satisfy the general Ricci and Bianchi identities.
Commutation Relations:
- The n-th order covariant derivative on a smooth manifold with affine connection is a differential operator that maps a function to a tensor field of type (0, n). The properties of this operator related to the permutations of indices are explored using non-associative algebra, and a general formula for these commutation relations is provided.

Conclusion

Covariant derivatives are essential tools in differential geometry, allowing for the differentiation of tensor fields in a manner consistent with the geometric structure of the space. They can be interpreted through group theory and extended to act on a wide range of geometric quantities, simplifying tensor analysis in both flat and curved spaces. Additionally, they play a crucial role in the study of eigenfunctions on Riemannian manifolds and satisfy important identities in the context of general connections.

What is the meaning of covariant derivatives?

Summary

Introduction

Key Insights

Conclusion

Sources and full results

A note on the meaning of covariant derivatives in supersymmetry

Covariant derivatives in supersymmetry are horizontal left-invariant vector fields on superspace, obtained from a (super)Lie group with a principal bundle and canonical connection.

Extension of covariant derivative (I): From component form to objective form

The generalized covariant derivative simplifies tensor analysis in curved coordinate systems by acting on any geometric quantity, including base vectors, vectors, and tensors.

Extension of covariant derivative (II): From flat space to curved space

The generalized covariant derivative simplifies tensor analysis on curved surfaces and allows for construction of differential and integral invariabilities.

Covariant derivatives of eigenfunctions along parallel tensors over space forms

The covariant derivative of an eigenfunction for the Laplace-Beltrami operator on a Riemannian manifold is a scalar multiple of the eigenfunction, with a polynomial depending on the eigenvalue.

General covariant derivatives for general connections

General covariant derivatives of vertical-valued tensor fields satisfy the general Ricci and Bianchi identities for general connections on a fibered manifold and a classical connection on the base.

Commutation relations on the covariant derivative

The n-th order covariant derivative on a smooth manifold with affine connection has properties related to permutations of indices, and the general formula for commutation relations is obtained.

Covariant time derivatives for dynamical systems

Covariant time derivatives are essential for formulating physical laws in a frame-independent manner, and can be derived using arbitrary time-dependent coordinates.

Quantum covariant derivative: a tool for deriving adiabatic perturbation theory to all orders

The quantum covariant derivative is a versatile tool for calculating nonlinear response properties in quantum systems driven by a Hamiltonian H=H(x) depending on slowly-varying parameters.

The Covariant Surface Derivative

Surface covariant derivatives do not commute and are not metrinilic with respect to the covariant basis S, leading to the curvature tensor.

Covariant Derivative in Gauge Theory

Gauge covariant derivative plays a crucial role in local gauge invariance and has applications in sound propagation in continuous fluids and solids.

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