are essential tools in Riemannian geometry and general relativity. They enable us to differentiate vector fields, define , and construct geodesics on manifolds. These concepts form the foundation for understanding intrinsic geometry and its relation to curvature and torsion.
By providing a coordinate-independent way to compare vectors at different points, affine connections allow us to explore the shape and structure of manifolds. They play a crucial role in various areas of mathematics and physics, from differential geometry to general relativity and beyond.
Affine connections
Play a crucial role in the study of Riemannian geometry and general relativity by providing a way to differentiate vector fields and define parallel transport
Allow for the comparison of vectors at different points on a manifold and the construction of geodesics, which are the "straightest possible" curves on a manifold
Form the foundation for understanding the intrinsic geometry of a manifold and how it relates to the curvature and torsion of the space
Motivation for affine connections
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Need for a notion of differentiation on manifolds that is independent of the choice of coordinates
Desire to compare vectors at different points on a manifold and define parallel transport
Importance in understanding the intrinsic geometry of a manifold and its relation to curvature and torsion
Definition of affine connection
An on a manifold M is a map ∇:X(M)×X(M)→X(M) satisfying certain properties
∇fXY=f∇XY for all smooth functions f and vector fields X,Y
∇X(fY)=f∇XY+X(f)Y for all smooth functions f and vector fields X,Y
∇X(Y+Z)=∇XY+∇XZ for all vector fields X,Y,Z
Provides a way to differentiate vector fields on a manifold
Covariant derivative
The of a vector field Y with respect to another vector field X is denoted by ∇XY
Measures the rate of change of Y along the direction of X
Satisfies the properties of an affine connection
Allows for the comparison of vectors at different points on a manifold
Parallel transport
Parallel transport is the process of moving a vector along a curve while preserving its "direction" with respect to the affine connection
A vector field Y is parallel along a curve γ if ∇γ˙Y=0, where γ˙ is the tangent vector to the curve
Parallel transport allows for the comparison of vectors at different points on a manifold
Depends on the choice of affine connection
Geodesics
Geodesics are curves that parallel transport their own tangent vector
Mathematically, a curve γ is a geodesic if ∇γ˙γ˙=0
Geodesics are the "straightest possible" curves on a manifold with respect to the affine connection
Play a crucial role in understanding the geometry of a manifold (shortest paths between points)
Torsion tensor
The T of an affine connection ∇ is defined as T(X,Y)=∇XY−∇YX−[X,Y]
Measures the failure of the connection to be symmetric
An affine connection is called torsion-free if T=0
Torsion plays a role in understanding the geometry of a manifold and its relation to the affine connection
Curvature tensor
The R of an affine connection ∇ is defined as R(X,Y)Z=∇X∇YZ−∇Y∇XZ−∇[X,Y]Z
Measures the failure of parallel transport to be path-independent
Provides information about the intrinsic curvature of the manifold
Plays a crucial role in understanding the geometry of a manifold and its relation to the affine connection (Riemann tensor in general relativity)
Levi-Civita connection
The is a unique torsion-free affine connection that is compatible with a given Riemannian metric
Defined by the : 2⟨∇XY,Z⟩=X⟨Y,Z⟩+Y⟨Z,X⟩−Z⟨X,Y⟩−⟨X,[Y,Z]⟩+⟨Y,[Z,X]⟩+⟨Z,[X,Y]⟩
Provides a natural way to differentiate vector fields on a Riemannian manifold
Used to define geodesics and parallel transport in Riemannian geometry
Affine connections vs metric connections
An affine connection is a more general concept than a metric connection
A metric connection is an affine connection that is compatible with a given metric (Levi-Civita connection)
Not all affine connections are metric connections, but all metric connections are affine connections
The choice of affine connection can have significant implications for the geometry of a manifold
Affine connections in local coordinates
In local coordinates, an affine connection is characterized by its
The Christoffel symbols Γijk are defined by ∇∂i∂j=Γijk∂k
The Christoffel symbols determine how the basis vectors change as one moves along a curve
The Christoffel symbols can be used to compute the covariant derivative, parallel transport, and geodesics in local coordinates
Christoffel symbols
The Christoffel symbols Γijk are the components of an affine connection in local coordinates
Symmetric in the lower indices if and only if the connection is torsion-free
For the Levi-Civita connection, the Christoffel symbols can be expressed in terms of the metric tensor: Γijk=21gkl(∂igjl+∂jgil−∂lgij)
Play a crucial role in computing the covariant derivative, parallel transport, and geodesics in local coordinates
Affine connections on vector bundles
The concept of an affine connection can be generalized to vector bundles over a manifold
A connection on a vector bundle E is a map ∇:X(M)×Γ(E)→Γ(E) satisfying certain properties
∇fXs=f∇Xs for all smooth functions f, vector fields X, and sections s of E
∇X(fs)=f∇Xs+X(f)s for all smooth functions f, vector fields X, and sections s of E
∇X(s+t)=∇Xs+∇Xt for all vector fields X and sections s,t of E
Allows for the differentiation of sections of a vector bundle along vector fields on the base manifold
Affine connections and Riemannian metrics
On a Riemannian manifold (M,g), there is a unique torsion-free affine connection compatible with the metric, called the Levi-Civita connection
The Levi-Civita connection provides a natural way to differentiate vector fields and define parallel transport on a Riemannian manifold
The curvature tensor of the Levi-Civita connection is closely related to the , which encodes information about the intrinsic geometry of the manifold
The study of is central to understanding the geometry of Riemannian manifolds
Affine connections and Lie groups
On a Lie group G, there is a natural affine connection called the
The Maurer-Cartan connection is defined using the left-invariant vector fields on G
The Maurer-Cartan connection is flat (has zero curvature) and torsion-free
The study of affine connections on Lie groups is important in understanding the geometry of homogeneous spaces and symmetric spaces
Affine connections in general relativity
In general relativity, spacetime is modeled as a 4-dimensional Lorentzian manifold (M,g)
The Levi-Civita connection of the spacetime metric plays a crucial role in the formulation of the theory
The geodesics of the Levi-Civita connection represent the paths of freely falling particles in the absence of external forces
The curvature tensor of the Levi-Civita connection, known as the Riemann tensor, encodes information about the gravitational field and the geometry of spacetime
The study of affine connections in general relativity is essential for understanding the nature of gravity and the structure of spacetime
Key Terms to Review (14)
Affine connection: An affine connection is a mathematical structure that allows for the comparison of vectors at different points in a manifold. It defines how to 'connect' tangent spaces and provides a way to differentiate vector fields along curves. This concept is essential for understanding how to compute the covariant derivative, establish the Levi-Civita connection, and analyze properties of geodesics through Jacobi fields and variations.
Affine connections: Affine connections are mathematical structures that define how vectors in a manifold can be parallel transported and how to differentiate vector fields along curves. In the context of general relativity, affine connections play a crucial role in describing how spacetime is curved by mass and energy, as well as providing a means to define geodesics, which represent the paths of free-falling particles.
Affine connections and Riemannian metrics: An affine connection is a way of specifying how to transport vectors along curves on a manifold, allowing for the definition of parallel transport and covariant derivatives. Riemannian metrics, on the other hand, provide a means to measure distances and angles on a manifold, enabling the study of geometric properties. Together, they form the foundation of differential geometry, allowing for a deeper understanding of curvature and geometric structures on manifolds.
Affine connections on vector bundles: Affine connections on vector bundles are mathematical structures that allow for the definition of parallel transport and covariant differentiation in the context of a vector bundle over a manifold. They provide a way to compare vectors in different fibers of the bundle, enabling the study of how vector fields behave along curves in the base manifold.
Affine connections vs Metric connections: Affine connections are mathematical objects that define how to differentiate vectors along a manifold, allowing for parallel transport and the definition of geodesics. In contrast, metric connections incorporate a notion of distance and angles through a Riemannian or pseudo-Riemannian metric, which enables the measurement of lengths and angles between vectors in addition to defining curvature. Both types of connections serve essential roles in differential geometry but focus on different aspects of geometry.
Christoffel symbols: Christoffel symbols are mathematical objects used in differential geometry to describe how coordinates change in a curved space. They play a critical role in defining connections and curvature on manifolds, enabling the calculation of geodesics, covariant derivatives, and the Levi-Civita connection.
Covariant Derivative: The covariant derivative is a way of specifying a derivative that accounts for the curvature of a manifold, allowing for the differentiation of vector fields along curves in a way that respects the geometric structure of the space. It plays a crucial role in understanding how quantities change as you move along surfaces and relates directly to other essential concepts such as connections, curvature, and geodesics.
Curvature Tensor: The curvature tensor is a mathematical object that measures the curvature of a Riemannian manifold, capturing how much the geometry of the manifold deviates from being flat. It relates to various fundamental concepts, such as geodesics, lengths, volumes, and the behavior of curves within the manifold, providing crucial insights into the geometric structure and its implications on physics, particularly in general relativity.
Koszul formula: The Koszul formula provides a way to compute the covariant derivative of a tensor product of vector fields in differential geometry. It expresses how the covariant derivative operates on the product of two vector fields, allowing us to relate the derivatives of each field to their Lie bracket and the connection coefficients. This is crucial for understanding the structure of connections and how they behave in curved spaces.
Levi-Civita connection: The Levi-Civita connection is a unique affine connection on a Riemannian manifold that preserves the metric and is torsion-free. This connection plays a central role in defining the covariant derivative, which allows for the differentiation of vector fields along curves in a way that respects the manifold's geometric structure.
Maurer-Cartan Connection: The Maurer-Cartan connection is a specific type of connection defined on the tangent bundle of a Lie group, which encodes how to differentiate along curves in the group. It provides a way to relate the algebraic structure of the Lie group to the geometric structure of its tangent space, allowing us to understand how elements and their infinitesimal variations behave under group operations. This connection is essential for studying properties such as curvature and holonomy in the context of differential geometry.
Parallel Transport: Parallel transport is a method of moving vectors along a curve on a manifold while keeping them 'parallel' according to the manifold's connection. This process is crucial for understanding how vectors behave in curved spaces, and it ties into various concepts like induced metrics on submanifolds, covariant derivatives, and geodesics, helping to maintain the geometric structure of the manifold.
Riemann curvature tensor: The Riemann curvature tensor is a fundamental mathematical object that measures the intrinsic curvature of a Riemannian manifold. It encapsulates how much the geometry of the manifold deviates from being flat and is crucial for understanding the behavior of geodesics, the nature of parallel transport, and the geometric properties of the space.
Torsion Tensor: The torsion tensor is a mathematical object that measures the failure of a connection to be symmetric in its lower two indices. It captures the twisting of a manifold and is crucial for understanding the geometric properties of spaces equipped with an affine connection. Torsion arises when the order of covariant differentiation matters, affecting the path taken between points on a manifold, and it relates closely to concepts of curvature and parallel transport.