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This function on the set of unit tangent vectors is often also called the Ricci curvature, since knowing it is equivalent to knowing the Ricci curvature tensor. The Ricci curvature is determined by the sectional curvatures of a Riemannian manifold, but generally contains less information.
[a] [1] [2] [3] It is also the modern name for what used to be called the absolute differential calculus (the foundation of tensor calculus), tensor calculus or tensor analysis developed by Gregorio Ricci-Curbastro in 1887–1896, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900. [4]
Since the Ricci tensor of a Riemannian metric also assigns to each p a symmetric bilinear form on T p M, the following definition is meaningful. Given a smooth manifold M and an open real interval (a, b), a Ricci flow assigns, to each t in the interval (a,b), a Riemannian metric g t on M such that ∂ / ∂t g t = −2 Ric g t.
The Weyl tensor has the same basic symmetries as the Riemann tensor, but its 'analogue' of the Ricci tensor is zero: = = = = The Ricci tensor, the Einstein tensor, and the traceless Ricci tensor are symmetric 2-tensors:
In general relativity and tensor calculus, the contracted Bianchi identities are: [1] = where is the Ricci tensor, the scalar curvature, and indicates covariant differentiation.
With this convention, the Ricci tensor is a (0,2)-tensor field defined by R jk =g il R ijkl and the scalar curvature is defined by R=g jk R jk. (Note that this is the less common sign convention for the Ricci tensor; it is more standard to define it by contracting either the first and third or the second and fourth indices, which yields a Ricci ...
The Ricci tensor: R σν, comes from the need in Einstein's theory for a curvature tensor with only 2 indices. It is obtained by averaging certain portions of the Riemann curvature tensor. The scalar curvature: R, the simplest measure of curvature, assigns a single scalar value to each point in a space. It is obtained by averaging the Ricci tensor.
In Ricci's notation, he refers to "systems" with covariant and contravariant components, which are known as tensor fields in the modern sense. [16] In the 20th century, the subject came to be known as tensor analysis, and achieved broader acceptance with the introduction of Albert Einstein's theory of general relativity, around 1915. General ...