Riemann curvature tensor — Safekipedia Discoverer

The Riemann curvature tensor is a key idea in a branch of mathematics called differential geometry. It helps us understand how space is curved. Named after mathematicians Bernhard Riemann and Elwin Bruno Christoffel, this tool assigns a special kind of number, called a tensor, to each point in a space. These numbers show how the space bends or curves at that point.

In simple terms, the Riemann curvature tensor measures how things change when you look at them from different directions in a curved space. If a space has no curvature, it is flat and looks like the ordinary Euclidean space we are used to. But if the space is curved, this tensor tells us exactly how and where the curvature occurs.

This idea is very important in physics, especially in the theory of general relativity. General relativity explains gravity not as a force, but as the curvature of spacetime caused by mass and energy. The Riemann curvature tensor helps scientists describe how this curvature affects the motion of objects, such as the way Earth orbits the Sun. It even helps explain the tidal forces we feel from the pull of the Moon and Sun on Earth’s oceans.

Definition

The Riemann curvature tensor is a way to describe how space curves in math. It helps us understand the shape of space by looking at how certain calculations change when we move from one point to another.

This tensor is important because it shows how the rules for moving in space break when the space is not flat, like the surface of a sphere. It is used in studying shapes and spaces in advanced math.

Main article: Riemann curvature tensor

Geometric meaning

Imagine you're walking on a flat tennis court with a tennis racket held out in front of you. As you walk around the court in a loop, keeping the racket pointed the same way, it will still point the same way when you return to your starting point. This is because the tennis court is flat.

Figure showing the geometric meaning of the Riemann curvature tensor in a spherical curved manifold. The fact that this transfer can define two different arrows at the starting point gives rise to the Riemann curvature tensor. The orthogonal symbol indicates that the dot product (provided by the metric tensor) between the transmitted arrows (or the tangent arrows on the curve) is zero. The angle between the two arrows is zero when the space is flat and greater than zero when the space is curved. The more curved the space, the greater the angle.

Now imagine walking on the curved surface of the Earth. Start at the equator, pointing your racket north. Walk to the north pole, then sideways, then back down to the equator, and finally back to your starting point. When you return, your racket will no longer point north—it will point west instead. This happens because the Earth's surface is curved. The Riemann curvature tensor helps us measure this kind of curvature in mathematics.

The Riemann curvature tensor shows how much a shape curves by looking at how vectors change when moved along paths. It tells us how the rules for moving in straight lines change on curved surfaces.

Coordinate expression

When we talk about how space curves, we can use special math tools called tensors. The Riemann curvature tensor is one of these tools. It helps us understand the shape of space at any point by looking at how tiny changes in direction affect each other.

We can write this tensor using something called coordinate vector fields and another tool named Christoffel symbols. These help us break down the tensor into smaller parts that are easier to work with in calculations. This way, mathematicians can study the curves and bends in space more clearly.

Main article: List of formulas in Riemannian geometry

Symmetries and identities

The Riemann curvature tensor follows special rules, called symmetries and identities, that help mathematicians study curved spaces. These rules tell us how the tensor behaves under different operations and calculations.

One important rule is known as the Bianchi identity, named after the mathematician who discovered it. This identity helps us understand how the curvature changes as we move through the space. These symmetries and identities are key to exploring the properties of curved surfaces and higher-dimensional spaces in mathematics.

Main article: Bianchi identity

Skew symmetry	R ( u , v ) = − R ( v , u ) {\displaystyle R(u,v)=-R(v,u)}	R a b c d = − R a b d c ⇔ R a b ( c d ) = 0 {\displaystyle R_{abcd}=-R_{abdc}\Leftrightarrow R_{ab(cd)}=0}
Skew symmetry	⟨ R ( u , v ) w , z ⟩ = − ⟨ R ( u , v ) z , w ⟩ {\displaystyle \langle R(u,v)w,z\rangle =-\langle R(u,v)z,w\rangle }	R a b c d = − R b a c d ⇔ R ( a b ) c d = 0 {\displaystyle R_{abcd}=-R_{bacd}\Leftrightarrow R_{(ab)cd}=0}
First (algebraic) Bianchi identity	R ( u , v ) w + R ( v , w ) u + R ( w , u ) v = 0 {\displaystyle R(u,v)w+R(v,w)u+R(w,u)v=0}	R a b c d + R a c d b + R a d b c = 0 ⇔ R a [ b c d ] = 0 {\displaystyle R_{abcd}+R_{acdb}+R_{adbc}=0\Leftrightarrow R_{a[bcd]}=0}
Interchange symmetry	⟨ R ( u , v ) w , z ⟩ = ⟨ R ( w , z ) u , v ⟩ {\displaystyle \langle R(u,v)w,z\rangle =\langle R(w,z)u,v\rangle }	R a b c d = R c d a b {\displaystyle R_{abcd}=R_{cdab}}
Second (differential) Bianchi identity	( ∇ u R ) ( v , w ) + ( ∇ v R ) ( w , u ) + ( ∇ w R ) ( u , v ) = 0 {\displaystyle \left(\nabla _{u}R\right)(v,w)+\left(\nabla _{v}R\right)(w,u)+\left(\nabla _{w}R\right)(u,v)=0}	R a b c d ; e + R a b d e ; c + R a b e c ; d = 0 ⇔ R a b [ c d ; e ] = 0 {\displaystyle R_{abcd;e}+R_{abde;c}+R_{abec;d}=0\Leftrightarrow R_{ab[cd;e]}=0}

Ricci curvature

The Ricci curvature tensor is a simpler way to understand the shape of space, created by combining parts of the Riemann tensor. It helps us study how space curves and bends in different directions.

Special cases

For a flat surface, like a piece of paper, the Riemann tensor is very simple because it has only one important piece of information. This piece of information is called the Gaussian curvature, and it tells us how curved the surface is at each point.

A space form is a special kind of curved space where the curvature is the same everywhere. In these spaces, the Riemann tensor has a very neat and simple form, depending only on this constant curvature. This shows how the shape of the space affects how things move and bend within it.

Main article: Space form