Curvatures in Riemannian geometry measure how a manifold deviates from being flat. measures the Gaussian curvature of geodesic planes, averages sectional curvatures, and provides a single value for total curvature at each point.
These curvatures are interconnected and offer complementary information about a manifold's geometry. They have intuitive interpretations: sectional curvature controls geodesic spreading, Ricci curvature affects volume growth, and scalar curvature relates to large-scale volume behavior and .
Definitions of curvatures
Curvatures are fundamental invariants in Riemannian geometry that measure the deviation of a manifold from being flat
They provide important information about the intrinsic geometry and shape of a Riemannian manifold
Sectional curvature
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Measures the Gaussian curvature of geodesic planes in a Riemannian manifold
Determined by the Riemannian R(X,Y,Z,W) evaluated on pairs of orthonormal vectors X,Y
Formula: K(X,Y)=∣X∧Y∣2R(X,Y,Y,X)
Provides the most complete description of curvature at a point
Ricci curvature
Obtained by taking the trace of the Riemannian curvature tensor
Measures the average sectional curvature in each tangent direction
Formula: Ric(X,Y)=∑i=1nR(X,ei,ei,Y) for an orthonormal basis {ei}
Captures the amount of volume distortion in the manifold
Scalar curvature
Further contraction of the Ricci curvature
Defined as the trace of the Ricci tensor
Formula: S=∑i=1nRic(ei,ei) for an orthonormal basis {ei}
Provides a single real number that represents the total curvature at each point
Relationships between curvatures
The different notions of curvature are interconnected and provide complementary information about the geometry of a Riemannian manifold
Ricci curvature vs sectional curvature
Ricci curvature is the average of sectional curvatures over all planes containing a given direction
If the sectional curvature is constant (space forms), then the Ricci curvature is proportional to the metric tensor
Positive (negative) Ricci curvature implies the existence of some positive (negative) sectional curvatures
Scalar curvature vs Ricci curvature
Scalar curvature is the trace of the Ricci tensor
Positive (negative) scalar curvature implies the existence of some positive (negative) Ricci curvatures
In dimension 2, the scalar curvature is twice the Gaussian curvature
The scalar curvature appears in the Hilbert-Einstein functional in general relativity
Geometric interpretations
The different curvatures have intuitive geometric interpretations that help understand their meaning
Sectional curvature interpretation
Measures the Gaussian curvature of geodesic planes
Controls the spreading or focusing of geodesics and the growth of
Positive (negative) sectional curvature implies that geodesics tend to converge (diverge)
Ricci curvature interpretation
Measures the average convergence or divergence of geodesics in each direction
Controls the volume growth of small balls in the manifold
Positive (negative) Ricci curvature implies that the volume of balls grows slower (faster) than in Euclidean space
Scalar curvature interpretation
Provides a rough measure of the total curvature at each point
Related to the asymptotic behavior of the volume of large balls
Appears in the Hilbert-Einstein functional, which is the action functional for general relativity
Computations and formulas
Computing the different curvatures involves the Riemannian curvature tensor and its contractions
Sectional curvature formula
K(X,Y)=∣X∧Y∣2R(X,Y,Y,X) for orthonormal vectors X,Y
Can be computed using the Christoffel symbols and their derivatives
In local coordinates: K(∂i,∂j)=Rijij
Ricci curvature formula
Ric(X,Y)=∑i=1nR(X,ei,ei,Y) for an orthonormal basis {ei}
In local coordinates: Ricij=∑k=1nRikjk
Can be expressed in terms of the Christoffel symbols and the metric tensor
Scalar curvature formula
S=∑i=1nRic(ei,ei) for an orthonormal basis {ei}
In local coordinates: S=gijRicij
Can be computed using the metric tensor and its derivatives
Curvature in special cases
Certain classes of Riemannian manifolds have special curvature properties that simplify their geometry
Curvature of surfaces
For 2-dimensional Riemannian manifolds (surfaces), the sectional curvature and Gaussian curvature coincide
The scalar curvature is twice the Gaussian curvature
Gauss's Theorema Egregium states that the Gaussian curvature is an intrinsic invariant
Curvature of space forms
Space forms are Riemannian manifolds with constant sectional curvature (spheres, Euclidean spaces, hyperbolic spaces)
The Ricci curvature is proportional to the metric tensor: Ric=(n−1)Kg
The scalar curvature is a constant multiple of the sectional curvature: S=n(n−1)K
Curvature of product manifolds
The sectional curvature of a product manifold (M1×M2,g1⊕g2) is determined by the curvatures of the factors
The Ricci curvature of a product manifold is the direct sum of the Ricci curvatures of the factors
The scalar curvature of a product manifold is the sum of the scalar curvatures of the factors
Curvature and topology
The curvature of a Riemannian manifold is closely related to its topology and can provide obstructions to the existence of certain geometric structures
Influence on Betti numbers
The Betti numbers bk(M) measure the rank of the homology groups Hk(M;R)
The Bochner technique relates the Ricci curvature to the vanishing of Betti numbers
Positive Ricci curvature implies the vanishing of Betti numbers in certain degrees
Influence on fundamental group
The fundamental group π1(M) captures the non-trivial loops in a manifold
The Cartan-Hadamard theorem states that a complete, simply connected manifold with non-positive sectional curvature is diffeomorphic to Rn
Negative Ricci curvature implies the fundamental group is infinite
Gauss-Bonnet theorem
Relates the total integral of the Gaussian curvature to the Euler characteristic χ(M) for compact surfaces
Formula: ∫MKdA=2πχ(M)
Generalizes to higher dimensions using the Pfaffian of the curvature form
Variational properties
Curvatures arise naturally in variational problems on the space of Riemannian metrics
Hilbert-Einstein functional
The Hilbert-Einstein functional is the action functional for general relativity
Defined as S(g)=∫M(Sg+Lm)dVg, where Lm is the Lagrangian density of matter
Critical points of the functional are Einstein metrics, which satisfy Ricg=λg for some constant λ
Yamabe problem
Aims to find a metric of constant scalar curvature in a given conformal class
Equivalent to solving the nonlinear PDE −Δgu+4(n−1)n−2Sgu=λun−2n+2
Solved affirmatively by Yamabe, Trudinger, Aubin, and Schoen
Ricci flow
A geometric evolution equation that deforms a in the direction of its Ricci curvature
Defined as ∂t∂g(t)=−2Ricg(t)
Used by Perelman to prove the Poincaré conjecture and Thurston's geometrization conjecture
Applications and examples
Curvature plays a central role in various areas of mathematics and physics
General relativity
In general relativity, spacetime is modeled as a 4-dimensional Lorentzian manifold
The relate the Ricci curvature to the energy-momentum tensor of matter
Solutions to the Einstein equations describe the geometry of spacetime in the presence of matter and energy
Comparison geometry
relate the curvature of a manifold to the geometry of model spaces (space forms)
The Rauch comparison theorem compares Jacobi fields on a manifold to those on a space form
The Bishop-Gromov volume comparison theorem compares the volume growth of balls to that in a space form
Sphere theorems
Sphere theorems provide sufficient conditions for a manifold to be homeomorphic or diffeomorphic to a sphere
The 1/4-pinched sphere theorem states that a complete, simply connected manifold with 41<K≤1 is homeomorphic to a sphere
The positive mass theorem in general relativity is a type of sphere theorem in Lorentzian geometry
Key Terms to Review (17)
Bernhard Riemann: Bernhard Riemann was a German mathematician whose work laid the foundation for Riemannian geometry and significantly advanced the study of differential geometry. His ideas are essential for understanding concepts like curvature, geodesics, and the mathematical properties of curved spaces, connecting various aspects of geometry to physics and other areas.
Comparison Geometry: Comparison geometry is a field of differential geometry that studies the geometric properties of spaces by comparing them to model spaces with known curvature properties. It allows mathematicians to draw conclusions about the curvature and topology of a given space by using comparison results with simpler, more understood geometric structures, such as spaces of constant curvature. This approach is particularly useful when analyzing sectional, Ricci, and scalar curvatures in various manifolds.
Comparison Theorems: Comparison theorems are essential results in differential geometry that allow the analysis of geometric properties of a space by comparing it to spaces of known curvature. They help establish relationships between geodesics, curvature, and topological features, providing a way to understand the behavior of manifolds through these comparisons.
Curvature Operator: The curvature operator is a mathematical object that encodes information about the curvature of a Riemannian manifold. It acts on the space of tangent vectors and produces a quadratic form that helps describe how curves bend within the manifold. This operator is essential for understanding various types of curvature, including sectional, Ricci, and scalar curvatures, as it provides a framework to analyze how the geometry of the manifold varies with respect to different directions.
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.
Einstein Field Equations: The Einstein Field Equations (EFE) are a set of ten interrelated differential equations that describe how matter and energy in the universe influence the curvature of spacetime. These equations are the cornerstone of general relativity, linking the geometry of spacetime to the distribution of mass and energy. The EFE have profound implications, particularly in understanding gravitational phenomena such as black holes and the dynamics of the universe as a whole.
Elie Cartan: Elie Cartan was a French mathematician renowned for his groundbreaking contributions to differential geometry and the theory of Lie groups. His work laid the foundation for understanding curvature, particularly through the formulation of the curvature tensor, which plays a crucial role in describing geometric properties of manifolds. Cartan's insights also advanced the study of symmetries in geometry, connecting these ideas with physical concepts such as relativity.
Gauss-Bonnet Theorem: The Gauss-Bonnet Theorem is a fundamental result in differential geometry that connects the geometry of a surface to its topology. Specifically, it states that for a compact two-dimensional Riemannian manifold, the integral of the Gaussian curvature over the surface is related to the Euler characteristic of the manifold, which is a topological invariant. This theorem reveals profound insights about the interplay between geometric properties, such as curvature, and topological features, like holes and surfaces.
General Relativity: General relativity is a fundamental theory of gravitation proposed by Albert Einstein, describing how matter and energy influence the curvature of spacetime. This theory extends the principles of special relativity and provides a framework to understand the dynamics of objects under gravitational influence, leading to key concepts in differential geometry such as curvature and geodesics.
Jacobi fields: Jacobi fields are vector fields along a geodesic that measure the variation of geodesics with respect to initial conditions. They play a crucial role in understanding the stability and behavior of geodesics, particularly in relation to conjugate points and the geometry of the manifold.
Local vs global curvature analysis: Local vs global curvature analysis refers to the examination of curvature properties of a manifold on two distinct scales. Local curvature focuses on the behavior of curves and surfaces in a small neighborhood around a point, while global curvature looks at the overall shape and structure of the manifold as a whole. Understanding both local and global aspects is crucial when studying various types of curvatures, such as sectional, Ricci, and scalar curvatures, as they provide insights into the geometric properties and potential applications in physics and other fields.
Negatively curved manifold: A negatively curved manifold is a type of geometric structure where the curvature at every point is less than zero, meaning it exhibits a hyperbolic geometry. This curvature causes the angles of triangles to sum to less than 180 degrees and allows for the existence of infinitely many parallel lines through a given point. This property relates closely to various types of curvature measures, revealing unique characteristics about the manifold's shape and behavior.
Positively Curved Manifold: A positively curved manifold is a type of geometric structure where every point has a curvature that is greater than zero, implying that it bends outward in all directions. This means that, locally, the geometry resembles that of a sphere. In the context of curvature measures, such as sectional, Ricci, and scalar curvatures, positive curvature influences the behavior of geodesics, the volume of balls within the manifold, and can lead to significant conclusions about the manifold's topology.
Ricci curvature: Ricci curvature is a mathematical concept that quantifies the degree to which the geometry of a Riemannian manifold deviates from being flat. It is derived from the Riemann curvature tensor and provides crucial information about the shape of the manifold, particularly in understanding volume and structure in relation to the presence of matter in general relativity.
Riemannian metric: A Riemannian metric is a smoothly varying positive definite inner product on the tangent spaces of a differentiable manifold, allowing one to measure distances and angles on that manifold. This concept forms the backbone of Riemannian geometry, enabling the exploration of curves, surfaces, and their properties in various contexts, from smooth manifolds to curvature concepts and beyond.
Scalar Curvature: Scalar curvature is a measure of the intrinsic curvature of a Riemannian manifold, reflecting how the geometry of the manifold deviates from being flat. It is derived from the Riemann curvature tensor and captures important geometric properties, connecting deeply with various other curvature concepts and providing insight into the manifold's shape and structure.
Sectional Curvature: Sectional curvature is a geometric concept that measures the curvature of a Riemannian manifold in two-dimensional sections spanned by tangent vectors. This curvature helps in understanding how geodesics behave in different directions and plays a crucial role in distinguishing various geometric properties of the manifold.