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Riemannian Manifold: A Natural Extension of Euclidean Space

Key Takeaways

  • Riemannian manifolds can also be defined as smooth manifolds with Riemannian metrics that define the shortest distance (otherwise called geodesics) from one point to another. 

  • Pseudo-Riemannian metrics and semi-Riemannian metrics are two classifications of Riemannian metrics.

  • Riemannian manifold is the natural extension of Euclidean space.

 Differential geometry

Differential geometry is the branch of mathematics that studies geometric properties using integral or differential calculus

Geometry is associated with angles, lengths, distances, curvature, areas, and volume. When solving geometric problems, differential calculus can be utilized, and this method corresponds to differential geometry.

Riemannian geometry, a branch of differential geometry, is the study of manifolds or space provided in Riemannian metrics. Riemannian geometry relates the local Riemannian manifold geometry to its global topology. Riemannian geometry, metrics, and manifolds are of great importance in the formulation of physics theories. In this article, we will take a look at differential geometry, Riemannian geometry, and the Riemannian manifold. 

Differential Geometry 

Differential geometry is the branch of mathematics that studies geometric properties using integral or differential calculus. Geometric properties are the properties that solely belong to the geometric object. One of the most common geometric properties is length. All geometric properties remain the same under congruence. The geometric properties that are preserved are congruent properties.

A knowledge of differential geometry is important when you study objects that are geometrically non-uniform. Conventional geometry studies objects made up of points, lines, planes, circles, etc. When dealing with objects that are not made of these shapes, it becomes difficult to analyze their properties. To determine the properties of things or objects made out of curves, different curvatures, or without corners, differential geometry is helpful.

Differential geometry takes into account the curves of non-constant curvature and helps in the study of curves, surfaces, and manifolds associated with an object. Differential geometry can provide information regarding what happens in a curve. Instead of pure geometric techniques and algebraic equations, differential geometry relies on differential calculus to solve geometric problems. Differential geometry brings methods of local and global analysis for solving geometric problems. 

Applications of Differential Geometry 

The impact of differential geometry is seen in representation theory analysis, group theory, statistics, algebraic, and differential topologies. There are extensive applications of differential geometry in Hamiltonian mechanics, medical imaging, geometric design, computer vision, geometrothermodynamics, information geometry, and geometric control. Most of the theories in physics can be formulated in a differential geometric framework. Some examples of physics theories employing differential geometric formulations are:

  1. In electromagnetism, the formulations can be in differential terms. One of the significant uses is in the generalized Stokes theorem for differential manifold. 

  2. Differential geometric formulations are used in quantum mechanics as well as quantum field theory.

  3. In classical Hamiltonian mechanics, the state-space (momentum + position) can be modeled as a manifold. 

  4. In general relativity theory, space-time can be modeled as a 4D-pseudo Riemannian manifold. 

Riemannian manifold is a smooth manifold endowed with Riemannian metrics. A new branch of geometry called Riemannian geometry is based on the idea of curvature and its relation to the topology. We will explore Riemannian geometry, Riemannian metrics, and Riemannian manifold in the upcoming sections. 

Riemannian Geometry 

Riemannian geometry is a generalized version of differential geometry. Riemannian geometry uses Riemannian metrics to study smooth manifolds. In solving curves, surfaces, and manifolds-related problems in Riemannian or differential geometry, calculus and linear algebra are used. 

Locally, the properties of the manifolds belong to Euclidean space or other topological spaces (in higher dimensions). For every Euclidean geometric concept, there is a natural analogue in Riemannian geometry, expressed in Riemannian metrics. For example, distance is expressed in a smooth positive definite bilinear form in Riemannian metrics. 

Riemannian Metrics 

In three-dimensional Euclidean space, the first fundamental form of a surface can be generalized under Riemannian metrics. The geometry of a space based on a definite Riemannian metric forms Riemannian geometry. Based on the generalization made on the concept of Riemannian metrics, there are different classifications of these metrics.

Pseudo-Riemannian metrics or space - When the metric is defined using non-definite, non-degenerate quadratic forms.

Semi-Riemannian metrics - When the determinant (gij) of the function gij(x) which defines the metric is zero. 

Riemannian Manifold

The term manifold represents an abstract mathematical space where the neighborhood of every point resembles Euclidean space. The Riemannian manifold can also be defined as smooth manifolds with Riemannian metrics that define the shortest distance (otherwise called geodesics) from one point to another. Riemannian manifold is geodesically complete, as there is no boundary or singular point in it that can be reached in a finite time. By measuring the curvature of the Riemannian manifold, the deviance from local flatness can be obtained. 

Riemannian manifold is the natural extension of Euclidean space. In the Riemannian manifold, there exists the re-interpretation for any standard operations present in Euclidean space. As global analysis and the theory of differential equations are important not only in mathematics and physics but also in engineering and technology, the Riemannian manifold is of great importance. 

Cadence’s suite of software can help you solve differential geometric problems. Subscribe to our newsletter for the latest CFD updates or browse Cadence’s suite of CFD software, including Fidelity and Fidelity Pointwise, to learn more about how Cadence has the solution for you. 

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