5.1 Gradient vector fields on manifolds

Gradient vector fields on manifolds are a key concept in Morse Theory. They're defined using Riemannian metrics and allow us to study how smooth functions behave on curved spaces. This connects calculus on flat spaces to more complex geometric settings.

Understanding gradient vector fields on manifolds is crucial for analyzing critical points and gradient flows. These tools help us explore the relationship between a manifold's shape and the behavior of functions defined on it, laying the groundwork for deeper insights in Morse Theory.

Gradient Vector Fields and Manifolds

Definition and Properties of Gradient Vector Fields

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• Gradient vector field defined as the vector field whose value at each point is the gradient of a scalar function at that point
• Gradient vector fields are conservative, meaning they have zero curl and the work done by the field along any closed path is zero
• Gradient vector fields are always perpendicular to the level sets of the scalar function they are derived from
• The integral of a gradient vector field along a curve depends only on the endpoints of the curve, not the path taken (path independence)

Manifolds and Tangent Spaces

• Manifold is a topological space that locally resembles Euclidean space near each point
  • Examples include curves, surfaces, and higher-dimensional spaces that can be described by a set of coordinates
• Tangent space at a point on a manifold is the vector space containing all possible tangent vectors to the manifold at that point
  • Tangent vectors represent the instantaneous directions in which a point can move along the manifold
• Tangent bundle is the disjoint union of all tangent spaces of a manifold, forming a new manifold

Riemannian Metrics and Gradient Vector Fields on Manifolds

• Riemannian metric is a smooth, positive definite, symmetric bilinear form on each tangent space of a manifold
  • Allows the computation of lengths, angles, and volumes on the manifold
  • Induces an inner product on each tangent space
• Gradient vector field of a smooth function on a Riemannian manifold is defined using the Riemannian metric
  • The gradient is the unique vector field such that the inner product of the gradient with any tangent vector equals the directional derivative of the function in the direction of the tangent vector

Derivatives and Smooth Functions

Covariant Derivatives and Parallel Transport

• Covariant derivative is a generalization of the directional derivative that accounts for the curvature of the manifold
  • Measures the change of a vector field along a curve on the manifold
• Parallel transport is the process of moving a tangent vector along a curve on the manifold while preserving its angle with the curve
  • Defined using the covariant derivative by requiring the covariant derivative of the vector field along the curve to be zero

Smooth Functions and Local Coordinates

• Smooth function on a manifold is a function that has continuous derivatives of all orders
  • Smoothness is a local property and can be checked using local coordinates
• Local coordinates are a set of functions that bijectively map an open subset of the manifold to an open subset of Euclidean space
  • Allow the manifold to be described locally using Euclidean coordinates
  • Smooth functions can be expressed in terms of local coordinates, and their derivatives can be computed using the chain rule

Gradient Vector Fields in Local Coordinates

• Gradient vector field of a smooth function can be expressed in terms of local coordinates
  • The components of the gradient in local coordinates are given by the partial derivatives of the function with respect to the coordinate functions
• The Riemannian metric can also be expressed in local coordinates as a symmetric, positive definite matrix
  • The gradient vector field can be computed by multiplying the inverse of the metric matrix with the vector of partial derivatives of the function

Critical Points and Gradient Flow

Critical Points and Their Classification

• Critical point of a smooth function on a manifold is a point where the gradient vector field vanishes
  • Examples include local minima, local maxima, and saddle points
• Critical points can be classified based on the behavior of the function in a neighborhood of the point
  • The Hessian matrix, which contains the second partial derivatives of the function, can be used to classify critical points (Morse Lemma)
    • If the Hessian is positive definite, the critical point is a local minimum
    • If the Hessian is negative definite, the critical point is a local maximum
    • If the Hessian has both positive and negative eigenvalues, the critical point is a saddle point

Gradient Flow and Morse Theory

• Gradient flow is the flow generated by the negative gradient vector field of a smooth function
  • Integral curves of the negative gradient vector field are called gradient flow lines
  • Gradient flow lines always flow from higher values of the function to lower values
• Morse theory studies the relationship between the critical points of a smooth function and the topology of the manifold
  • Morse functions are smooth functions whose critical points are non-degenerate (the Hessian is non-singular)
  • The Morse inequalities relate the number of critical points of each index (the number of negative eigenvalues of the Hessian) to the Betti numbers of the manifold
• Gradient flow can be used to prove the Morse inequalities and to study the attachment of handles to the manifold as the level sets of the function pass through critical points