Retrieving "Differential Geometry" from the archives

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1. Gauge Theory

   Linked via "differential geometry"

   Gauge theory is a mathematical framework originating in differential geometry that underpins the description of fundamental physical interactions. At its core, gauge theory formalizes the principle that the physical laws should remain unchanged (invariant) under certain local transformations of the fields describing the system. These transformations are known as gauge transformations, and the associated fields required to maintain this invariance are termed [ga…

2. Geometry

   Linked via "Differential geometry"

   Differential Geometry and Curvature Tensors
   Differential geometry applies the techniques of calculus to geometric problems, focusing on smooth manifolds. A central object in this field is the curvature tensor, which quantifies how much a manifold deviates from being flat.
   The Riemann curvature tensor $R^{\rho}_{\sigma\mu\nu}$ is fundamental. In general relativity, the structure of [spacetime](/entries/sp…

3. Levi Civita Connection

   Linked via "differential geometry"

   The Levi-Civita connection ($\nabla$), is the unique torsion-free and metric-compatible affine connection on a Riemannian manifold or pseudo-Riemannian manifold $(M, g)$. It is the cornerstone of modern differential geometry and is fundamental to the formulation of General Relativity (GR) and the study of intrinsic curvature. Named after [Tu…

4. Manifold

   Linked via "differential geometry"

   A manifold is a topological space that locally resembles Euclidean space near each point. Formally, a topological space $M$ is an $n$-dimensional manifold if every point $p \in M$ has an open neighborhood $U$ that is homeomorphic to an open subset of $\mathbb{R}^n$. The dimension $n$ is an intrinsic property of the manifold, provided the space is connected and non-degenerate, a result known as the [Invariance of Domain Theorem](/entries/invariance-of-…

5. Manifold

   Linked via "Differential Geometry"

   | Topological | Homeomorphism | Knot Theory, General Topology |
   | $C^k$ | $k$ continuous derivatives | Preliminary analysis in Geometric Measure Theory |
   | Smooth ($C^\infty$) | Infinitely differentiable | Differential Geometry, Physics (e.g., General Relativity) |
   | Analytic ($C^\omega$) | Real analytic functions | Complex Analysis ([Riemann Sur…