Glossary

3.2 Non-degenerate critical points and their classification

3 min read•august 7, 2024

Non-degenerate critical points are key in Morse theory. They have non-singular Hessian matrices, allowing us to classify them as local minima, maxima, or saddle points. This classification helps us understand a function's behavior near these points.

The lets us approximate smooth functions near non-degenerate critical points as quadratic forms. This approximation, along with the of the critical point, gives us crucial insights into the function's local topology and behavior.

Critical Points and Their Classification

Non-Degenerate Critical Points

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  • has a non-singular Hessian matrix at the point
  • Characterized by the determinant of the Hessian matrix being non-zero
  • Allows for the classification of critical points into three types:
  • Helps determine the local behavior of a function near the critical point

Local Extrema

  • Local minimum is a non- where the function value is less than or equal to nearby points
    • Hessian matrix at a local minimum has all positive eigenvalues
    • Example: the point (0,0)(0, 0) for the function f(x,y)=x2+y2f(x, y) = x^2 + y^2
  • Local maximum is a non-degenerate critical point where the function value is greater than or equal to nearby points
    • Hessian matrix at a local maximum has all negative eigenvalues
    • Example: the point (0,0)(0, 0) for the function f(x,y)=−x2−y2f(x, y) = -x^2 - y^2
  • Local extrema provide information about the function's behavior in a neighborhood around the critical point

Saddle Points

  • Saddle point is a non-degenerate critical point that is neither a local minimum nor a local maximum
  • Hessian matrix at a saddle point has both positive and negative eigenvalues
  • Example: the point (0,0)(0, 0) for the function f(x,y)=x2−y2f(x, y) = x^2 - y^2
  • Saddle points indicate a change in the function's behavior from increasing to decreasing (or vice versa) along different directions
  • The number of negative eigenvalues of the Hessian matrix at a saddle point is called the index of the saddle point

Morse Lemma and Quadratic Forms

Morse Lemma

  • Morse lemma states that near a non-degenerate critical point, a smooth function can be written as a quadratic form plus higher-order terms
  • Provides a local approximation of the function near the critical point
  • Allows for the classification of critical points based on the quadratic form
  • Helps in understanding the local topology of the function near the critical point

Quadratic Forms

  • Quadratic form is a homogeneous polynomial of degree 2 in n variables
  • Can be represented by a symmetric matrix (the Hessian matrix in the context of Morse theory)
  • The number of positive, negative, and zero eigenvalues of the matrix determines the type of the quadratic form:
    • Positive definite: all eigenvalues are positive
    • Negative definite: all eigenvalues are negative
    • Indefinite: both positive and negative eigenvalues
  • The type of the quadratic form corresponds to the type of the non-degenerate critical point (minimum, maximum, or saddle)

Index of a Critical Point

  • Index of a non-degenerate critical point is the number of negative eigenvalues of the Hessian matrix at that point
  • Corresponds to the number of independent directions in which the function decreases from the critical point
  • Local minimum has an index of 0, local maximum has an index equal to the dimension of the domain, and saddle points have an index between 0 and the dimension of the domain
  • The index helps classify critical points and provides information about the local topology of the function near the critical point

Key Terms to Review (19)

Degenerate Critical Point: A degenerate critical point is a point in a function where the gradient (or derivative) is zero, but the behavior of the function at that point does not exhibit the expected characteristics of typical critical points. Unlike non-degenerate critical points, which are associated with distinct curvature and can often indicate local maxima or minima, degenerate critical points can lead to more complex scenarios like saddle points or flat regions, requiring additional analysis to understand their significance.
Differential Forms: Differential forms are mathematical objects used in calculus on manifolds, generalizing the concepts of functions and differentials to higher dimensions. They play a crucial role in various areas, including integration on manifolds and the generalization of Stokes' theorem, which relates integrals over boundaries to integrals over the domains they enclose. Understanding differential forms is essential for working with tangent and cotangent spaces, classifying critical points, and exploring the relationship between topology and geometry.
Differential Topology: Differential topology is a branch of mathematics that focuses on the properties and structures of differentiable functions on differentiable manifolds. It connects analysis, topology, and geometry, providing tools to study smooth shapes and their deformations, especially in understanding critical points and their implications for manifold classification.
Gradient Flow: Gradient flow refers to the flow generated by following the negative gradient of a function, effectively describing how a system evolves over time towards its critical points. This concept is crucial in understanding the dynamics of functions, particularly in relation to their critical points, where local minima and maxima exist, and connects deeply with various topological and geometrical properties of manifolds.
Homology: Homology is a mathematical concept used to study topological spaces by associating algebraic structures, called homology groups, which capture information about the shape and connectivity of the space. This notion plays a vital role in understanding the properties of manifolds and CW complexes, as it relates to the classification of critical points and provides insights into cobordism theory.
Index: In the context of Morse Theory, the index refers to a critical point of a smooth function and is defined as the number of negative eigenvalues of the Hessian matrix at that point. This concept is crucial for classifying critical points and understanding the topology of the underlying manifold, as it provides insight into the local behavior of functions near these points. The index helps in distinguishing between different types of critical points, including minima, maxima, and saddle points, and plays a significant role in analyzing Morse functions and applying the Morse Lemma.
Index 0: Index 0 refers to a classification of critical points in the context of Morse theory, specifically denoting points where the Hessian matrix of a smooth function has exactly zero positive eigenvalues. This classification indicates that the critical point corresponds to a local minimum and plays a significant role in understanding the topology of manifolds by examining how functions behave near these points.
Index 1: In Morse Theory, the index 1 refers to a specific classification of non-degenerate critical points of a smooth function defined on a manifold. A critical point is termed non-degenerate if the Hessian matrix at that point is invertible, and the index corresponds to the number of negative eigenvalues of this Hessian. The significance of index 1 is that it characterizes saddle points, which are essential in understanding the topology of manifolds and the behavior of functions near critical points.
Index 2: Index 2 refers to a specific classification of critical points in the context of Morse Theory, indicating that the second derivative test at that point yields a signature of two negative eigenvalues and no positive eigenvalues. This classification is essential for understanding the topology of manifolds, as critical points with index 2 correspond to local maxima, which play a significant role in the overall behavior of a function on a manifold.
Local maximum: A local maximum refers to a point in a function where the value is greater than or equal to the values of the function at nearby points. This concept is crucial in understanding critical points, as it helps classify the behavior of functions and their extrema in various contexts such as differentiable functions, Morse theory, and gradient vector fields.
Local Minimum: A local minimum is a point in a function where the function's value is lower than that of its neighboring points, indicating that it is a relative low point in the surrounding area. Understanding local minima is crucial when analyzing critical points, as they help classify the behavior of functions, especially in the context of optimization and topological features.
Manifold Theory: Manifold theory studies mathematical spaces that locally resemble Euclidean space. These spaces, known as manifolds, are essential in various fields such as physics and engineering as they provide a way to understand complex shapes and structures. The classification of non-degenerate critical points plays a crucial role in understanding the topology of these manifolds, which ultimately influences the behavior of functions defined on them.
Morse Function: A Morse function is a smooth real-valued function defined on a manifold that has only non-degenerate critical points, where the Hessian matrix at each critical point is non-singular. These functions are crucial because they provide insights into the topology of manifolds, allowing the study of their structure and properties through the behavior of their critical points.
Morse Lemma: The Morse Lemma states that near a non-degenerate critical point of a smooth function, the function can be expressed as a quadratic form up to higher-order terms. This result allows us to understand the local structure of the function around critical points and connects deeply to various concepts in differential geometry and topology.
Morse-Sard Theorem: The Morse-Sard Theorem states that the set of critical values of a smooth function on a manifold has measure zero in the codomain. This powerful result implies that most values attained by the function are not critical, and it has profound implications for the study of critical points and their classifications, the topology of manifolds, and certain geometric transformations like sphere eversion.
Morse's Theorem: Morse's Theorem states that for a smooth function defined on a manifold, the topology of the manifold can be understood through the critical points of the function. This theorem connects the nature of these critical points—specifically non-degenerate ones—to the topology of the manifold by describing how these points lead to changes in the topology as one moves through different levels of the function.
Non-degenerate critical point: A non-degenerate critical point of a smooth function is a point where the gradient is zero, and the Hessian matrix at that point is invertible. This condition ensures that the critical point is not flat and allows for a clear classification into local minima, maxima, or saddle points, which connects to many important aspects of manifold theory and Morse theory.
Saddle Point: A saddle point is a type of critical point in a function where the point is neither a local maximum nor a local minimum. It is characterized by having different curvature properties along different axes, typically resulting in a configuration where some directions yield higher values while others yield lower values.
Topological Features: Topological features refer to the distinct characteristics and structures of a space that are preserved under continuous transformations, such as stretching or bending, but not tearing or gluing. In the context of critical points and their classification, understanding topological features is essential for analyzing how these points affect the shape and structure of the manifold or surface they belong to, revealing important insights about the underlying space's geometry and dynamics.
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