In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point. Specifically, the scalar curvature represents the amount by which the volume of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space. In two dimensions, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In more than two dimensions, however, the curvature of Riemannian manifolds involves more than one functionally independent quantity.

In general relativity, the scalar curvature is the Lagrangian density for the Einstein–Hilbert action. The Euler–Lagrange equations for this Lagrangian under variations in the metric constitute the vacuum Einstein field equations, and the stationary metrics are known as Einstein metrics. The scalar curvature is defined as the trace of the Ricci tensor, and it can be characterized as a multiple of the average of the sectional curvatures at a point. Unlike the Ricci tensor and sectional curvature, however, global results involving only the scalar curvature are extremely subtle and difficult. One of the few is the positive mass theorem of Richard Schoen, Shing-Tung Yau and Edward Witten. Another is the Yamabe problem, which seeks extremal metrics in a given conformal class for which the scalar curvature is constant.

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space. As such, it provides one way of measuring the degree to which the geometry determined by a given Riemannian metric might differ from that of ordinary Euclidean n-space. More generally, the Ricci tensor is defined on any pseudo-Riemannian manifold. Like the metric itself, the Ricci tensor is a symmetric bilinear form on the tangent space of the manifold.

The Ricci curvature is broadly applicable to modern Riemannian geometry and general relativity theory. In connection with the latter, it is up to an overall trace term, the portion of the Einstein field equation representing the geometry of spacetime, the other significant portion of which comes from the presence of matter and energy. In connection with the former, lower bounds on the Ricci tensor on a Riemannian manifold allow one to extract global geometric and topological information by comparison (cf. comparison theorem) with the geometry of a constant curvature space form. If the Ricci tensor satisfies the vacuum Einstein equation, then the manifold is an Einstein manifold, which have been extensively studied (cf. Besse 1987). In this connection, the Ricci flow equation governs the evolution of a given metric to an Einstein metric, the precise manner in which this occurs ultimately leads to the solution of the Poincaré conjecture.

In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds.

Definition. The sectional curvature of the plane spanned by the (linearly independent) tangent vectors of the Riemannian manifold is

.

In local coordinates, if

we then have

which implies

Besides

Thus

.

To be exact, without using Einstein summation convention, one reads the above identity as following

.

For a given smooth function on manifold , the gradient of is given by

.

Note that gradient of is also a vector field on . Thus, for each , it is reasonable to talk about .

Definition 1. Hessian of , denoted by , is defined as the symmetric -tensor

.

We also denote by the following

.

Thus

Note that

.

Since then

which implies

.

Definition 2. Laplacian of , denoted by , is defined as the trace of .

Note that is a -tensor, then in local coordinates, one has

.

It is clear that is a -tensor field. To see this fact, one can assume then from

one has

since which is exactly an -tensor. Then we can define divergence of a vector field as following

Definition 3. Divergence of vector field is given by

.

In coordinates, this is

and with respect to an orthornormal basis

.

Thus .

Recall that is defined to be

.

Let

.

The way to understand is to look at the following 4-covariant tensor

.

As can be seen, the components of are .

We first obtain the following result.

Theorem 1. The curvature tensor   satisfies the following property

.

Proof.

The proof relies on the definition of the 4-covariant tensor above. To be precise, one has

and

.

Since

then . This comes from the definition of curvature tensor and the fact that

.

Similarly, for the latter case, one can argue as follows

.

We now use the fact that is a metric connection. Indeed,

Thus

Hence

.

The above identity also holds if we replace by a vector field . Thus

which implies

.

Therefore, .

Corollary 1. and .

Theorem 2 (the first Bianchi identity). The curvature tensor   satisfies the following property

.

Proof. Since

then

.

Similarly,

.

Since is torsion free, one gets

.

As a consequence,

.

Now

which implies

.

If we change by , by we then obtain

which implies, by using Theorem 1,

.

Corollary 2. .

Corollary 3. Followed from the proof of Theorem 2, by pairing with to the both sides one has

.

Theorem 3 . The curvature tensor   satisfies the following property

.

Proof. By the first Bianchi indentity,

which implies

.

Thus

.

Similarly, by changing , , and one gets

.

Hence by using Theorem 1.

Theorem 4 (the second Bianchi identity). The curvature tensor   satisfies the following property

.

Proof. One can use the normal coordinates in order to simplify the calculation. Indeed, normal coordinates tell us at a given point that

and

for all , , . Thus,

which implies

Hence

which implies

.

Similarly, we can write down and . Summing up we get the desired result.

Suppose is a differentiable manifold of dimension $n$.

Connection on vector bundles

Definition 1. A connection on a vector bundle is a map

which satisfies the following conditions

• For any , .
• For and any , .

If is a tangent vector field on (i.e. a section of the tangent bundle ) one can define a covariant derivative along , denoted by , as follows

where represents the pairing between and .

Locally, a connection is given by a set of differential 1-forms. Suppose is a coordinate neighborhood of with local coordinates , . Choose smooth sections of on such that they are linearly independent everywhere. Such a set of sections is called a local frame field of on . It is obvious that at every point

forms a basis for the tensor space . Because is a local section on , we can write

where are smooth functions on . Denote then .

Definition 2 (curvature operator). Suppose are two arbitrary smooth tangent vector fields on the manifold . Then

is the curvature operator of the connection .

Obviously, has the following properties

• ,
• ,
• ,

where , and .

Connection on tangent bundles (affine connections)

A tangent bundle is an -dimensional vector bundle determined intrinsically by the differentiable structure of an -dimensional smooth manifold . A connection of is called an affine connection on . Affine connection is usually denoted by .

Definition 3 (curvature tensor). The curvature tensor is a (1,3)-tensor defined by

.

In local coordinates, the curvature tensor is given by

.

A simple calculation shows us that

.

Definition 4 (torsion tensor). The torsion tensor is a (1,2)-tensor defined by

.

In local coordinates, the torsion tensor is given by

.

A simple calculation shows us that

.

Definition 5 (torsion free). If the torsion tensor of an affine connection is zero, then the connection is said to be torsion free.

When is a Riemannian manifold with metric then we have the following definition

Definition 6 (Levi-Civita connection). An affine connection is called a Levi-Civita connection if:

• It preserves the metric, i.e., for any vector fields , , we have

  

  where denotes the derivative of the function along the vector field .

• It is torsion free.

The first condition above is called metric connection condition. Thus, the Levi-Civita connection is the torsion free metric connection, i.e., the torsion free connection on the tangent bundle (an affine connection) preserving a given Riemannian metric.

There is a theorem in the literature saying that the Levi-Civita connection is unique and it is given by the following identity

.

In local coordinate, the Levi-Civita connection is given by

where are called Christoffel symbols which are determined by

where .

Let’s start with a differentiable manifold M of dimension . Throughout this topic, we denote by a point on and its local chart (at ). A point is determined by hence it is often identified with . We usually denote by the local coordinates of .

Definition 1. A tangent vector at is a map defined on the set of the differentiable functions in a neighborhood of , where satisfies the following conditions

• is linear, that is to say: if , then .
• if is flat at , i.e. at .
• .

Definition 2. The tangent space at is the set of tangent vectors at .

From the definition 1, let us show that the tangent space of definition 2 has a natural vector space structure of dimension . We set

and .

With this sum and this product, is a vector space. And now let us exhibit a basis. It is reasonable to define the tangent vector at . Precisely,

Definition 3. The tangent vector at is defined to be

.

The vectors are independent and they form a basis for . We usually call the directional derivative of in the direction . For an arbitrary vector , one can define the directional derivative of in the direction as following

where denotes the -th component of in this coordinate chart.

We now assume further that is a Riemannian manifold where is its metric. We are now in a position to define gradient for a smooth function.

Definition 4. For any smooth function on a Riemannian manifold , the gradient of is the vector field such that for any vector field ,

,   i.e.  

where denotes the inner product of tangent vectors at  defined by the metric .

We now express the local form of the gradient at . By definition 4, one has in the local coordinates

.

If we assume we then have

.

Thus

which implies after multiplying both sides by the matrix

.

Therefore,

.

We end this topic by showing what is? Roughly speaking, at a point since is a vector, then is nothing but its magnitude. To be exact, one defines

.

In Riemannian geometry, the lower index means differentiation and the upper index means component, therefore, we usually use to denote the quantity . With this notation, .

of in the direction

In this topic, adapted from a paper due to Li Ma and Xingwang Xu published in Comptes Rendus Mathematique we shall give a non-existence result concerning the following Lichnerowicz equation in

, on

where , , and are given smooth functions of . To be precise, we obtain the following

Theorem. Suppose , , and . Let . Assume that

Then there exists no positive solution to the above Lichnerowicz equation.

Let us denote the integral

by . We call the average of on the sphere of radius , or sphere mean of a function around the origin.

Proof. Note that a simple calculation shows us that

.

Therefore

.

Since on the sphere , is also the outer normal vector, therefore

Thus by the divergence theorem, one gets

.

Hence

.

Differentiating once more yields

.

Since

then

.

Thus

Therefore, taking this average operation we have

.

Since for each fixed ,

Then by using the general Cauchy inequality, one gets

.

Thus,

.

It turns out that

,

which implies that

after an integration. This is because, by definition of the sphere mean,

Dividing both sides by and integrating this inequality over , we have

.

Sending we have

,

which is impossible by our assumption. The proof is complete.

Let us consider a probability space , i.e., is a measurable space together with . We assume further that are such that . Then we conclude that .

Indeed, since then . We write in the following way

.

We then see that since . Similarly, . Hence, .

In mathematics, absolute continuity is a smoothness property which is stricter than continuity and uniform continuity.

Definition. A finite function on a finite interval is said to be absolute continuous if and only if for given , there exists such that

for any collection (finite or not) of non-overlapping subintervals of with .

Statement. Show that is absolutely continuous on if and only if given , there exists such that

for any finite collection of non-overlapping subintervals of with .

Proof. If is absolutely continuous on , then the result is easily obtained by using the definition and the fact that for every .

Now we prove that

for given , there exists such that for any finite collection of non-overlapping subintervals of with .

Indeed, we split the collection into two types:

• type are all such that and
• type are all such that .

For given , there exists such that for any finite collection with . Then for we also have

.

Similarly,

.

From the following inequality with we deduce that

with the fact that

Note that the right hand side of the above inequality is bounded from above by

Thus, we have proved that for given , there exists such that

for any finite collection of non-overlapping subintervals of with . Letting we can claim that $f$ is absolutely continuous.