On the geometry of Riemannian manifolds with density
We introduce a new geometric approach to a manifold equipped with a smooth density function that takes a torsion-free affine connection, as opposed to a weighted measure or Laplacian, as the fundamental object of study. The connection motivates new versions of the volume and Laplacian comparison theorems that are valid for the -Bakry-Émery Ricci tensor, a weaker assumption than has previously been considered in the literature. As applications we prove new generalizations of Myers’ theorem and Cheng’s diameter rigidity result. We also investigate the holonomy groups of the weighted connection. We show that they are more general than the Riemannian holonomy, but also exhibit some of the same structure. For example, we obtain a generalization of the de Rham splitting theorem as well as new rigidity phenomena for parallel vector fields. A general feature of all of our rigidity results is that warped or twisted product splittings are characterized, as opposed to the usual isometric products.
Key words and phrases:
Ricci curvature for a Riemannian manifold equipped with a smooth positive density function was first considered by Lichnerowicz [Lichnerowicz70, Lichnerowicz72] and was systematically studied and vastly generalized by Bakry-Émery [BE] and their collaborators. Their approach is to study a weighted Laplacian and the curvature is defined to provide a Bochner formula for this operator acting on functions. This is also often viewed as the study of a manifold with smooth measure where the measure is defined so that the weighted Laplacian is self-adjoint. This approach has been extraordinarily fruitful as the Bakry-Émery Ricci tensors have become a fundamental concept in probability, analysis, and geometry and are important in Ricci flow, optimal transport, isoperimetric problems, and general relativity. In fact, Bakry-Émery’s work is vastly more general, as they make sense of Bochner formulas for a much larger class of operators. Given the reach of their work, it is remarkable that their ideas in the very special case of manifolds with density has had so many applications.
With the many applications of weighted Ricci curvature, it is also desirable to generalize other aspects of Riemannian geometry to the weighted setting. A standard introductory course in Riemannian geometry shows how the subject flows naturally from the existence of the unique Levi-Civita torsion free and compatible affine connection. In this paper we give a new approach to manifolds with density which also takes a torsion free affine connection as the fundamental object.
Let the Levi-Civita connection of a Riemannian manifold . For a one-form define . is a torsion free affine connection, moreover it is projectively equivalent to , meaning that has the same geodesics, up to re-parametrization, as . In fact, a result of Weyl [Weyl] states that any torsion-free connection projectively equivalent to is of the form for some . Any connection has a well defined -curvature tensor and -Ricci tensor given by the formulae
To see the link to Bakry-Émery Ricci curvature, let be a positive density function on M. The -Bakry-Émery Ricci tensor is defined as
where is a constant that is also allowed to be infinite, in which case we write . For a manifold of dimension larger than 1, if we let , then we recover a Bakry-Émery Ricci tensor, as a simple calculation (see Proposition 3.3) shows that
In other words, the Ricci tensor of a projectively equivalent connection is exactly the -Bakry-Émery Ricci tensor.
The Bakry-Emery Ricci tensor has traditionally been studied for values of the parameter , and this is where bounds on Bakry-Emery Ricci tensors are equivalent to the curvature dimension condition as defined by Bakry-Émery. There are many recent papers on this condition, some that are most relevant to the results of this paper are [Lott, Morgan, Morgan3, MunteanuWang, WeiWylie]. Also see chapter 18 of [MorganBook] and the references there-in.
Note that a lower bound on is a weaker condition than for . There is an emerging body of research on lower bounds on when . The first papers investigating the case [Ohta, KolesnikovMilman] appeared almost simultaneously. In [Ohta], Ohta extends results involving optimal transport and lower bounds on when along with extending the Bochner inequality, eigenvalue estimates, and the Brunn-Minkowski inequality. In [KolesnikovMilman] Kolesnikov-Milman also extend the Poincare and Brunn-Minkowski inequality for manifolds with boundary when . Milman also extends the Heintze-Karcher Theorem, isoperimetric inequality, and functional inequalities for in [Milman1]. In [Ohta2] it is also established that lower bounds on , are equivalent to curvature-dimension inequalities as defined by Lott-Vilanni [LV] and Sturm [Sturm1, Sturm2]. Also see [Klartag, Milman2, OhtaTakatsu1, OhtaTakatsu2]. For earlier related work for measures on Euclidean space see [Borell, BrascampLieb]. The first results for were proven by first author who showed generalizations of the splitting theorem [WylieSplitting] and Myers’ theorem [WylieSec], we will discuss these results below. The definition of Ricci curvature for singular torsion-free affine connections has also been investigated recently by Lott [Lott2].
The connection also recovers a notion of weighted sectional curvature that was introduced by the first author. Namely, if we let be the dual vector field to coming from the metric , then
This curvature was called in [WylieSec] and was further studied in [KennardWylie]. The study of weighted sectional curvature from the perspective of the connection will be the subject of a different paper.
The fact the curvature quantities and come from the connection not only gives motivation for their study, but also introduces a number of new tools. For example, the connection gives a preferred re-parametrization of geodesics, which in turn can be used to define new global “re-parametrized distance” function which we show plays a similar role to the distance function for comparison geometry theorems. We also show that, when is closed and is orientable, admits a parallel volume form. This gives a natural measure which is a slightly different from the weighted measure usually used in the study of Bakry-Émery Ricci curvature. Combining this re-parametrized volume with the re-parametrized distance gives generalizations of the volume and Laplacian comparison theorems to from which we obtain a new Myers’ theorem along with a new diameter rigidity result. The connection also gives a weighted concept of parallelism encoded in its holonomy groups. We show these weighted holonomy groups are more general than the Riemannian ones, but also admit some similar structural properties, such as a generalization of the de Rham splitting theorem.
In the next section we define these notions and state our main results in terms of these objects. Section 3 examines the basic properties of the connection . Section 4 contains various comparison principles for manifolds with density. Section 5 contains a study of the holonomy group of as well as general discussion of parallel tensors.
2. Definitions and statement of results
2.1. Re-parametrized distance
In this section we assume is a closed form and write . The connection gives rise to a re-parametrization of the geodesics. We normalize these reparametrized geodesics in the following way.
is a normalized -geodesic if
is a re-parametrization of a minimizing unit speed Riemannian geodesic , and
For points define the “re-parametrized distance” between and as the infimum of the time it takes to travel from to along a normalized -geodesic. That is,
where the infimum is taken over all normalized -geodesics . Assuming the metric is complete, is clearly finite and well-defined from basic properties of Riemannian geodesics. Let . If is not a cut point to , then there is a unique minimal geodesic from to and is smooth in a neighborhood of as can be computed by pulling the function back by the exponential map at . Note that and is zero if and only if and . However, does not define a metric since it does not satisfy the triangle inequality.
There is also a new natural normalization of the curvature coming from the re-parametrized geodesics. Namely, for a normalized -geodesic we have so,
Our first result is the following generalization of Myers’ theorem involving renormalized distance and the curvature bound.
Theorem 2.2 (Weighted Myers’ Theorem).
Let , , be a complete Riemannian manifold and let be a closed one-form, . Suppose that there is such that then for all
A corollary of the classical Myers’ theorem is that the manifold must be compact. In contrast, it is possible for to be uniformly bounded on a non-compact complete Riemannian manifold. Recall that the connection is called geodesically complete if all of the -geodesics can be extended for all time. If we additionally assume completeness of the connection we obtain the following natural corollary.
Let , , be a complete Riemannian manifold and let be a closed one-form such that is geodesically complete. If then is compact.
Since bounded implies is complete, Theorem 2.2 also recovers [WylieSec]*Theorem 1.6 which states that if and is bounded then is compact.
The completeness of where implies -completeness as defined in [WylieSplitting] and [WoolgarWylie] (See Propositon 3.4 below). As such, the generalization of the Cheeger-Gromoll splitting theorem proven in [WylieSplitting] can also be re-phrased in terms of the connection .
[WylieSplitting]*Theorem 6.3 Let , , be a complete Riemannian manifold and let be a closed one-form such that is geodesically complete. If and contains a line, then splits as a warped product.
The function is also naturally related to a conformal change of metric. Let then is the smallest length in the metric of a minimal geodesic between and in the metric. As such, . So Theorem 2.2 tells us that the diameter of the metric is less than or equal to . For this conformal diameter estimate we also obtain the following rigidity characterization.
Suppose that , , is complete and satisfies , . If there are points and such that then is a rotationally symmetric metric on the sphere.
Recall that Cheng [Cheng] showed that a complete Riemannian manifold satisfies and if and only if is a round sphere. We also give a complete characterization of the spaces satisfying the hypotheses of Theorem 2.6 and there are examples which are not constant curvature. Theorem 2.6 can thus be thought of as the analog in positive curvature to Theorem 2.5. It would be interesting to know whether this result is true under the weaker condition that . The main difficulty is that does not satisfy the triangle inequality.
The re-parametrized distance also has meaning for a negative lower bound on . In fact, the results above follow from a generalization of the Laplacian comparison theorem for the weighted Laplacian which is true for any constant , see Theorem 4.4 below.
2.2. Volume Comparison
Another insight coming from the connection is a natural volume comparison theory. In [WeiWylie] volume comparison theory was developed for the -volume, , for space satisfying and is bounded. We extend this theory to the weaker condition and bounded. We state only the special case here of the absolute volume comparison for . See Theorem 4.5 and Corollary 4.7 below for the general statements.
Suppose that , , satisfies and let be the minimum of on then
where is the volume of the ball in -dimensional Euclidean space. In particular,
where be the maximum of on
Yang showed that if and is bounded, then has polynomial growth of degree at most [Yang]. Theorem 2.7 extends this result to , and improves it, even in the case, to only require a lower bound on . We also obtain the same topological application that, if and is bounded below, then and any finitely generated subgroup of the fundamental group is polynomial growth of degree at most .
The connection also provides a new approach to volume comparison theory which yields a sharp relative volume comparison that assumes no a priori bounds on . This comes from the following observation.
Let be an orientable Riemannian manifold with Levi-Civita connection and smooth one-form . The connection admits a parallel volume form if and only if is closed. Moreover, if and the form is parallel with respect to .
For , define . Proposition 2.8 indicates that we should consider the measure instead of . In fact, we will see below that the same local estimates can be used to either give bounds on or . The measure arises if we change coordinates using the parametrized distance instead of the Riemannian distance function. As such, our volume comparison for will be in terms of the level sets of the re-parametrized distance instead of the metric balls. See Theorem 4.5 for the precise statement of this sharp relative volume comparison.
As an application of the volume comparison theorem, we obtain the following absolute volume comparison in the case .
Suppose that is a complete Riemannian manifold with supporting a function such that for some then and is finite.
2.3. Weighted Holonomy
The connection also introduces a new concept of parallelism for manifolds with measure. Recall that the holonomy group of a manifold equipped with a connection is the group of linear maps of the tangent space given by parallel translation around loops with a fixed base point. While Levi-Civita connections are characterized as the torsion free connections which have holonomy contained in , on an orientable manifold with closed 1-form , the holonomy of the connection is only contained, in general, in (see Proposition 5.2). This is natural if we consider the connection as a structure for a measure instead of a metric. While the holonomy groups of are more general than the Riemannian ones, we show they also exhibit some similar rigidity phenomena.
Recall the de Rham decomposition theorem which states that a Levi-Civita connection admits a parallel field if and only if the metric locally splits off a flat factor and, more generally, that the holonomy is reducible if and only if the metric is locally a product. We give examples showing these results are not true for the connection . However, the holonomy of does exhibit similar rigidity phenomena.
The spaces in our rigidity results will be warped or twisted products instead of direct products. Here by a twisted product we mean a Riemannian manifold which is a topological product with metric of the form where and are fixed metrics on and respectively and is an arbitrary positive function on . is a warped product if, in addition, is a function depending on only. First we state the result for parallel fields.
Let be a closed -form. If is complete and simply connected and admits linearly independent -parallel vector fields, then splits as one of the following:
In both cases, splits as (or ), so can also be thought of as a warped product.
In the more general case of reducible holonomy groups we obtain a twisted product splitting.
Theorem 2.12 (Weighted de Rham decomposition theorem).
A Riemannian metric admits a closed one-form such that the holonomy is reducible if and only if is locally isometric to a twisted product with . Moreover, if a compact manifold admits a closed one-form such that the holonomy is reducible, then the universal cover is diffeomorphic to and the covering metric is isometric to .
A consequence of the Riemannian de Rham theorem is that if the Riemannian holonomy is reducible then the holonomy group decomposes as a product. This is not true for the holonomy of as the splitting of the group will only be block upper triangular in general for a twisted product (see Example 5.7).
Comparison geometry results for the weighted Ricci and sectional curvatures like the ones above prove that many topological obstructions to Riemannian metrics with curvature bounds extend to the weighted curvatures. An open question is whether the topologies that support positive curvature are the same. Namely, given a triple with positive weighted Ricci or sectional curvatures it is an open question whether there is always some other metric on with positive Ricci or sectional curvature. See [WylieSplitting, KennardWylie] for further discussion.
The holonomy of the connection is also related to this question by the following result.
Suppose that supports a 1-form such that the holonomy group of is compact. Then,
If for all orthonormal pairs , , then there is a metric on with .
If then there is a metric on with .
Both parts also hold for non-negative, negative and non-positive curvature.
While we construct examples below showing that the holonomy group of need not be compact, Theorem 2.15 motivates the continued study of the holonomy of in relation to the study of weighted curvature bounds. For further discussion of the condition of compact holonomy, see subsection 5.4.
3. Connection for Manifolds with measure
Let be a Riemannian manifold with smooth one form . In this section we collect some basic facts about the weighted connection, . depends not only on but on as well, however, since we will always think of the background metric as being fixed, we will not emphasize this dependency.
It is easy to see that is a torsion free connection. Any linear connection defines a notion of geodesics as the curves whose velocity fields are parallel along the curve. Two connections are called projectively equivalent if they have the same geodesics up to parametrization. We call a curve an -geodesic if it is a geodesic for the connection . We will refer to the usual geodesics for the Levi-Civita connection as the -geodesics. By a theorem of Weyl, is projectively equivalent to . For completeness we verify this fact for when is closed, and also fix the re-parametrization that we will utilize for comparison results.
If is an -geodesic then the image of is a -geodesic, and the parametrization satisfies for some constant .
Let be an -geodesic, then
We conclude that . Dividing by we get a log derivative, which we solve to get , so . The image of the geodesic is the same, since is parallel to . ∎
Recall that the Levi-Civita connection has the universal property that it is the unique torsion free connection which is compatible with the metric. The next proposition shows that the weighted connection has a similar universal property for a smooth manifolds equipped with a smooth measure .
Given a Riemannian metric and a smooth measure there is a unique torsion free linear connection which is projectively equivalent to the Levi-Civita connection with respect to which is parallel. Moreover, if , then the connection is where .
We use Weyl’s theorem that any projectively equivalent connection is of the form . We can also use the fact that the Riemannian volume form is parallel with respect to the Levi-Civita connection. Then, for linearly independent fields we have
Therefore, is parallel if and only if . ∎
Now we turn our attention to the curvature of .
The curvature tensor of , is
whenever are orthonormal.
We compute the curvature tensor:
Finally, we get:
This also yields (1) and (2). ∎
To make our results in the next section easier to compare to other results for Bakry-Emery Ricci tensors, we will use the function so that . As mentioned in the introduction, this give the formula for the Ricci tensor, . It also gives us the measure .
Recall that a manifold with a connection is called geodesically complete if every geodesic is defined for all time. We say that , with , is -complete if is a geodesically complete connection. The next proposition shows that -completeness implies -completeness as defined in [WylieSplitting, WoolgarWylie]. Here denotes the reparametrized distance function as defined in Section 2.1
Let be a complete and -complete manifold, closed, then implies .
Suppose that there exists a sequence such that , but does not go to infinity. By passing to a subsequence, we can assume that for some fixed . Let be unit speed -geodesics with and such that . subconverges to a unit vector . Let be the -geodesic with and .
By uniform convergence of geodesics, for all . Therefore, , so the -geodesic with image is only defined up to some finite time. ∎
4. Comparison Principles
4.1. Volume element comparison
Now we consider comparison geometry results for on a Riemannian manifold of dimension . In this section we will consider all of our formulas in terms of the function .
Let and let , , be exponential polar coordinates (for the metric ) around which are defined on a maximal star shaped domain in called the segment domain. Write the volume element in these coordinates as .
Let be the reparametrized distance function defined in section 2.1 above. Inside the segment domain, has the simple formula
Therefore, is a smooth function in the segment domain with the property that . We can then also take to be coordinates which are also valid for the entire segment domain. We can not control the derivatives of in directions tangent to the geodesic sphere, so the new coordinates are not orthogonal as is the case for geodesic polar coordinates. However, this is not an issue when computing volumes as
Given a minimal unit speed geodesic with , the connection gives a natural re-parametrization of in terms of the function , we denote the derivative in the radial direction in terms of this parameter by . In geodesic polar coordinates has the expression We note it is not the same as in coordinates.
Recall that for a Riemannian manifold , where is the Riemannian Laplacian of the distance function to the point . (4.1) indicates we should consider the quantity
We thus recover the usual drift Laplacian considered by Bakry-Émery. Letting we find that satisfies a familiar differential inequality in terms of the parameter .
Let be a unit speed minimal geodesic with . Let be the parameter and let . Then
Moreover, if equality is achieved at a point then at that point has at most one non-zero eigenvalue which is of multiplicity
This is essentially Lemma 3.1 of [WylieSplitting]. We repeat the outline proof here for completeness. Begin with the usual Bochner formula for functions, which says that for any function ,
The Bochner formula for the -Laplacian and -Bakry-Émery Ricci curvature is given by
Consider this equation with at an interior point of a minimizing geodesic (so that is smooth in a neighborhood). Then , so the left hand side is zero. As is a null vector for , has at most non-zero eigenvalues and by Cauchy-Schwarz, . This gives us the equation along ,
Using this equation and completing a square gives us
Since , we have the desired equation in terms of . Moreover, equality in this inequality is achieved only if equality is achieved in Cauchy-Schwarz, which is equivalent to having at most one non-zero eigenvalue which is of multiplicity . ∎
Assuming the curvature bound we have the usual Riccati inequality
with the caveat that it is in terms of the parameter instead of . Define be the solution to , and , where prime denotes derivative with respect to . Define so that
Then solves This gives us the following comparison estimate.
Suppose be a manifold with density such that . Let , , and be defined as in Lemma 4.1. Then
where, when , we assume that .
Consider the quantity . Then
So we have . Since the only thing we need to show to show that is that is bounded. To see this note that
Where in the last line we have used the fact that . ∎
This estimate gives us the following estimate for the volume element
Suppose satisfies . Let be a point in and let be the volume element in geodesic polar coordinates then the function is non-increasing along any minimal geodesic with with .
4.2. The case
Due to our normalization of the function , the results of the previous section are only valid for a manifold of dimension . The reader might find it illuminating to consider the one dimensional case with where we get ordinary differential inequalities that are similar to the equations above. To avoid further technicalities we will just consider the ODE we get when these inequalities are equalities. Seeing the arguments in this simpler case may be helpful to the reader, but this section can also be skipped.
Since there is no Ricci curvature in dimension , we have where denotes the derivative with respect to a parameter . We consider the equation . Letting , this becomes
Define and the parameter such that . Then we have
This first order Riccati equation can be solved for . For simplicity, we assume or . Then we obtain
On the other hand, we have