Mathematical Physics, Analysis and Geometry - Volume 7

343 Pages·2004·2.44 MB·English

by  H. McKean (Editors)

Mathematical Physics, Analysis and Geometry 7: 1–8, 2004.
1
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Relating Thomas–Whitehead Projective
Connections by a Gauge Transformation
CRAIG ROBERTS
Department of Mathematics, Southeast Missouri State University, Cape Girardeau,
MO 63701-4799, U.S.A. e-mail:
2 CRAIG ROBERTS
element in the one-dimensional de Rham cohomology vector space of the base
manifold.
2. Structurally Equivalent TW-connections
The relevant definitions and results for TW-connections will be reviewed in this
section. Further details may be obtained by referring to (Roberts, 1992) and
(Roberts, 1995).
The construction of the principal R-bundle begins with a real n-dimensional
vector space V . If v is a nonzero element in the nth exterior product of V , then
the set ε = {±v} will be called a volume element. Defining a volume element
only up to sign allows for the consideration of both orientable and nonorientable
manifolds. The set of all volume elements will be denoted by E(V ). If M is a
smooth n-dimensional manifold and p an element of M, replacing V with TpM,
the tangent space of M at p, allows the bundle of volume elements over M, denoted
by E(M), to be defined as
⋃
E(M) = E(TpM).
p∈M
E(M) is the total space of a principal R-bundle over M with the projection map
π: E(M) → M defined by π(ε) = p. The structure group is the reals R under
a
addition, and the right action on E(M) is given by ε · a = e ε. The fundamental
vector field on E(M) corresponding to the element d/dt in the Lie algebra of R
will be called the canonical fundamental vector field on E(M) and denoted by ξ .
DEFINITION 1. A Thomas–Whitehead projective connection, or TW-connection,
is a torsionfree linear connection ∇ on E(M) that is invariant with respect to the
right action of R on E(M) and for which
1
∇ξ = − (id),
n + 1
where id is the identity (1, 1)-tensor and ξ the canonical fundamental vector field
on E(M).
A TW-connection ∇ induces a projective structure on the base manifold M in
˜ ˜
the following way. If ω is a connection 1-form on E(M) and X and Y are the
respective ω-horizontal lifts of smooth vector fields X and Y on M, then
ω
˜
∇
XY = π∗(∇X˜Y)
ω
defines a torsionfree linear connection ∇ on M. For the same TW-connection ∇,
ω
but a different connection 1-form ω, the torsionfree linear connection ∇ belongs
ω ω
to the same projective structure as ∇ . Furthermore, the mapping ω →↦ ∇ is
a one-to-one correspondence between the connection 1-forms on E(M) and the
GAUGE-RELATED TW-CONNECTIONS 3
ω
torsionfree linear connections belonging to the same projective structure as ∇ .
Consequently, the TW-connection ∇ induces a projective structure on the base
manifold M.
DEFINITION 2. TW-connections that induce the same projective structure on the
base manifold M are structurally equivalent.
Definition 2 is a change in terminology from (Roberts, 1992) and
(Roberts, 1995) designed to distinguish this equivalence class of TW-connections
from those that follow.
If the base manifold M has dimension one, then all TW-connections are struc-
turally equivalent in this special case. This follows since the induced connec-
tions on M all have the same geodesics up to a reparameterization, which im-
plies the induced connections belong to the same projective structure on M. See
(Spivak, 1979) and (Kobayashi, 1972).
Structurally equivalent TW-connections are more generally characterized by the
existence of a symmetric (0, 2)-tensor on E(M).
THEOREM 1. The TW-connections ∇ and ∇ are structurally equivalent if and
only if there is a unique symmetric (0, 2)-tensor β on E(M) such that the Lie
derivative Lξβ = 0, and
∇ = ∇ + (ιξβ) ⊗ id + id ⊗ (ιξβ) − β ⊗ ξ,
where (ιξ β) denotes a 1-form on E(M) defined by (ιξβ)(X) = β(ξ,X), for any
smooth vector field X on E(M), and satisfies (ιξβ)(ξ) = 0.
3. Ricci Equivalent TW-connections
DEFINITION 3. The Ricci tensor of a TW-connection ∇ is a (0, 2)-tensor on
E(M) defined by
Ricci(X, Y ) = trace{V →↦ R(V,X)Y },
where R is the curvature tensor of ∇ and X, Y , and V are smooth vector fields
on E(M).
The Ricci tensor of a TW-connection satisfies two special properties. First, since
a TW-connection is invariant with respect to the right action of R on E(M), its Ricci
tensor will be invariant with respect to this right action. Also, the Ricci tensor is
0 if one of the arguments is the canonical fundamental vector field ξ . These two
∗
properties imply that Ricci = π α, for a (0, 2)-tensor α on M.
For structurally equivalent TW-connections, a straightforward, but rather
lengthy, calculation shows the difference of their Ricci tensors may be expressed
in terms of the symmetric (0, 2)-tensor β from Theorem 1.
4 CRAIG ROBERTS
THEOREM 2. If ∇ and ∇ are structurally equivalent TW-connections and their
respective Ricci tensors are denoted by Ricci and Ricci, then
Ricci − Ricci = −(n + 1) d(ιξβ) +
( )
1
+ (n − 1) (ιξβ) ⊗ (ιξβ) − sym∇(ιξβ) + β ,
n + 1
where β and ιξβ are defined as in Theorem 1 and sym∇(ιξβ) denotes the symmet-
ric part of ∇(ιξβ).
DEFINITION 4. The TW-connections ∇ and ∇ are Ricci equivalent if they are
structurally equivalent and have identically equal Ricci tensors.
As was mentioned in the previous section, for the special case in which the base
manifold M has dimension one, all TW-connections are structurally equivalent. In
addition, whenever M has dimension one, E(M) has dimension two and the Ricci
tensor of any TW-connection is identically 0. Consequently, all TW-connections
are Ricci equivalent for the special case in which the base manifold has dimension
one.
More generally, Ricci equivalent TW-connections are characterized by sharp-
ening Theorem 1 to the following.
THEOREM 3. The TW-connections ∇ and ∇ are Ricci equivalent if and only if
there is a unique closed 1-form φ on E(M) such that φ(ξ) = 0 and
∇ = ∇ + φ ⊗ id + id ⊗ φ − (n + 1)(∇φ − φ ⊗ φ) ⊗ ξ.
Proof. If ∇ and ∇ are Ricci equivalent TW-connections, they are structurally
equivalent and Theorem 1 implies there exists a unique symmetric (0, 2)-tensor β
on E(M) such that Lξβ = 0, (ιξ β)(ξ) = 0, and
∇ = ∇ + (ιξβ) ⊗ id + id ⊗ (ιξβ) − β ⊗ ξ.
Since ∇ and ∇ also have identically equal Ricci tensors, the equation for the
difference of their Ricci tensors given by Theorem 2 becomes
( )
1
−(n + 1)d(ιξβ) + (n − 1) (ιξβ) ⊗ (ιξβ) − sym∇(ιξβ) + β = 0.
n + 1
Considering the skew-symmetric and symmetric parts, respectively, of this equa-
tion yields d(ιξβ) = 0 and
1
(ιξβ) ⊗ (ιξβ) − sym∇(ιξβ) + β = 0.
n + 1
From these equations, as well as ∇(ιξβ) = sym∇(ιξβ) − d(ιξβ), it follows that
β = (n+1)(∇(ιξβ)−(ιξβ)⊗(ιξβ)). Thus, setting φ = ιξβ yields a unique closed
1-form such that φ(ξ) = 0 and
∇ = ∇ + φ ⊗ id + id ⊗ φ − (n + 1)(∇φ − φ ⊗ φ) ⊗ ξ.
GAUGE-RELATED TW-CONNECTIONS 5
Conversely, if φ is a closed 1-form on E(M) and φ(ξ) = 0, then ∇φ =
sym∇φ − dφ = sym∇φ and (Lξφ)(X) = d(φ(ξ))(X) + 2 dφ(ξ,X) = 0, for
all smooth vector fields X on E(M). Furthermore, if φ is the unique closed 1-form
on E(M) satisfying
∇ = ∇ + φ ⊗ id + id ⊗ φ − (n + 1)(∇φ − φ ⊗ φ) ⊗ ξ,
then β = (n+ 1)(∇φ −φ ⊗φ) defines a unique symmetric (0, 2)-tensor on E(M)
for which Lξβ = 0, ιξβ = φ, and
∇ = ∇ + (ιξβ) ⊗ id + id ⊗ (ιξβ) − β ⊗ ξ.
Hence, ∇ and ∇ are structurally equivalent by Theorem 1. Substituting the expres-
sions for β and ιξβ into the equation for the difference in the Ricci tensors of ∇
and ∇ given by Theorem 2 and simplifying shows ∇ and ∇ have identical Ricci
tensors. Therefore, ∇ and ∇ are Ricci equivalent. ✷
4. Gauge Equivalent TW-connections
DEFINITION 5. A diffeomorphism g: E(M) → E(M) such that g(ε · a) =
g(ε) · a, for all ε in E(M) and real numbers a, and which induces the identity map
on the base manifold M is a gauge transformation of E(M).
For a TW-connection ∇, a gauge transformation g of E(M), and smooth vector
fields X and Y on E(M), the expression
−1
g
∗ (∇g∗(X)g∗(Y ))
defines a TW-connection since g commutes with the right action of R on E(M).
Also, a one-to-one correspondence between the group of gauge transformations
of E(M) and the set of smooth maps from M to R can be established by setting
g(ε) = ε · (f ◦ π)(ε), for all ε in E(M), where f is a smooth map from M to R
and π is the projection map from E(M) to M (Bleecker, 1981).
DEFINITION 6. The TW-connections ∇ and ∇ are gauge equivalent if there is a
gauge transformation g of E(M) such that
−1
∇XY = g
∗ (∇g∗(X)g∗(Y )),
for all smooth vector fields X and Y on E(M).
The next theorem shows gauge equivalence is a finer equivalence relation than
Ricci equivalence and structural equivalence.
THEOREM 4. Gauge equivalent TW-connections are Ricci equivalent.
6 CRAIG ROBERTS
Proof. If ∇ and ∇ are gauge equivalent TW-connections, there exists a gauge
−1
transformation g of E(M) such that ∇XY = g
∗ (∇g∗(X)g∗(Y )), for all smooth
vector fields X and Y on E(M). If R and R denote the respective curvature tensors
of ∇ and ∇, then
−1
R(X, Y )Z = g
∗ (R(g∗(X), g∗(Y ))g∗(Z)),
∗
where Z is a smooth vector field on E(M), and it follows that Ricci = g Ricci.
In the previous section, it was noted that there is a (0, 2)-tensor α on M such that
∗
Ricci = π α. Therefore,
∗ ∗ ∗ ∗ ∗
Ricci = g Ricci = g (π α) = (π ◦ g) α = π α = Ricci,
and the Ricci tensors of gauge equivalent TW-connections are identical.
It remains to show that the gauge equivalent TW-connections ∇ and ∇ are
structurally equivalent. Let ω be a connection form on E(M). For the gauge trans-
∗
formation g relating ∇ and ∇, g ω is also a connection form on E(M). Taking
∗
X and Y to be smooth vector fields on M, and denoting their respective g ω-
∗ ∗
horizontal lifts by X and Y , we have 0 = g ω(X) = ω(g∗(X)) and 0 = g ω(Y ) =
ω(g∗(Y )). Thus, the vector fields g∗(X) and g∗(Y ) on E(M) are the ω-horizontal
∗
ω g ω
lifts of X and Y , respectively. For the induced connections ∇ and ∇ on M, this
implies
ω −1
∇
XY = π∗(∇g∗(X)g∗(Y )) = (π ◦ g )∗(∇g∗(X)g∗(Y ))
∗
g ω
= π∗(∇
XY ) = ∇X Y
−1
since π ◦ g = π. Therefore, ∇ and ∇ induce the same projective structure on M
so they are structurally equivalent. ✷
Theorems 3 and 4 suggest the possibility of further refining Theorem 1 for
gauge equivalent TW-connections.
THEOREM 5. The TW-connections ∇ and ∇ are gauge equivalent if and only if
there is a unique exact 1-form φ on E(M) such that φ(ξ) = 0 and
∇ = ∇ + φ ⊗ id + id ⊗ φ − (n + 1)(∇φ − φ ⊗ φ) ⊗ ξ.
Proof. If ∇ and ∇ are gauge equivalent TW-connections, then Theorem 4 im-
plies they are Ricci equivalent. It must be shown that the unique closed 1-form
φ given by Theorem 3 is exact for the case of gauge equivalent TW-connections.
Recall from the proof of Theorem 3 that the property φ(ξ) = 0 for a closed 1-
form φ implies Lξφ = 0. Hence, there exists a closed 1-form ρ on M such that
∗
π ρ = φ, and the equation in Theorem 3 may be written as
∗ ∗ ∗ ∗ ∗
∇ = ∇ + π ρ ⊗ id + id ⊗ π ρ − (n + 1)(∇π ρ − π ρ ⊗ π ρ) ⊗ ξ.
GAUGE-RELATED TW-CONNECTIONS 7
˜
If ω is a connection form on E(M), X and Y smooth vector fields on M, and X and
˜
Y their respective ω-horizontal lifts, then this equation becomes
∗ ∗
˜ ˜ ˜ ˜ ˜ ˜
∇
X˜Y = ∇X˜Y + π ρ(X)Y + π ρ(Y )X −
∗ ∗ ∗
˜ ˜ ˜ ˜
−(n + 1)((∇π ρ)(X; Y ) − π ρ(X)π ρ(Y ))ξ.
ω ω
Projecting by π∗ to M gives ∇
XY = ∇XY +ρ(X)Y +ρ(Y )X. H. Weyl showed this
ω ω
equation implies ∇ and ∇ are projectively equivalent; in other words, they have
the same geodesics up to a reparameterization (Spivak, 1979). This is equivalent to
ω ω
∇ and ∇ belonging to the same projective structure on M (Kobayashi, 1972).
The gauge transformation g of E(M) relating ∇ and ∇ provides another means
ω ω
of obtaining the equation showing ∇ and ∇ are projectively equivalent. Since
g may be expressed as g(ε) = ε · (f ◦ π)(ε), for all ε in E(M) and some smooth
˜ ˜ ∗ ˜ ˜ ˜
map f from M to R, it follows that g∗(X) = X + π df (X)ξ and g∗(Y ) = Y +
∗ ˜ ˜ −1 ˜
π df (Y )ξ . Hence, ∇
X˜Y = g∗ (∇g∗(X˜)g∗(Y )) becomes
1 1
∗ ∗
˜ ˜ ˜ ˜ ˜ ˜
∇
X˜Y = ∇X˜Y − π df (X)Y − π df (Y )X +
n + 1 n + 1
1
∗ ∗ ∗ ∗
˜ ˜ ˜ ˜ ˜
+(X(π df (Y )) − π df (∇
X˜Y) + π df (X)π df (Y ))ξ.
n + 1
Projecting by π∗ to M gives
1 1
ω ω
∇ Y = ∇ Y − df(X)Y − df(Y )X,
X X
n + 1 n + 1
ω ω
which again shows ∇ and ∇ are projectively equivalent.
−1 ∗
Setting ρ = −(n+ 1) df shows ρ is exact, which in turn implies φ = π ρ is
exact.
Conversely, assume there exists an exact 1-form φ satisfying the hypotheses,
then ∇ and ∇ are Ricci equivalent by Theorem 3 since an exact 1-form is closed.
From the properties dφ = 0 and φ(ξ) = 0, it follows that Lξφ = 0, which implies
∗
there is a 1-form ρ on M such that φ = π ρ. Furthermore, ρ must be exact since
φ is an exact 1-form satisfying φ(ξ) = 0. Let f be the smooth map from M to R
∗
such that df = ρ. Hence, φ = π df . Define a gauge transformation g of E(M)
by g(ε) = ε · (−(n + 1)(f ◦ π)(ε)), for all ε in E(M). Thus, for smooth vector
fields X and Y on E(M),
−1 ∗ ∗
g
∗ (∇g∗(X)g∗(Y )) = ∇XY + π df(X)Y + π df(Y )X −
∗ ∗ ∗
− (n + 1)((∇π df)(X ; Y ) − π df(X)π df(Y ))ξ.
By assumption, the right-hand side of this equation is merely ∇XY since φ =
∗
π df . Therefore, ∇ and ∇ are gauge equivalent TW-connections. ✷
Since gauge equivalence is characterized by an exact 1-form and Ricci equiva-
lence is characterized by a closed 1-form, a relationship to the one-dimensional de
Rham cohomology vector space is suggested.
8 CRAIG ROBERTS
1
DEFINITION 7. The vector space H (M,R) = {closed 1-forms on M}⧸{exact
1-forms on M} is the one-dimensional de Rham cohomology vector space of M.
A pair of Ricci equivalent TW-connections ∇ and ∇ induce a de Rham coho-
mology class on M in a natural way. If φ is the unique closed 1-form on E(M)
given by Theorem 3, then recalling the proof of Theorem 5 there is a closed 1-form
∗
ρ on M such that φ = π ρ. The induced de Rham cohomology class on M is [ρ].
THEOREM 6. The TW-connections ∇ and ∇ are gauge equivalent if and only if
they are Ricci equivalent and the induced de Rham cohomology class on M is 0.
Proof. The result follows from Theorems 3 and 5 since the unique closed 1-form
φ characterizing a pair of Ricci equivalent TW-connections may be expressed as
∗
φ = π ρ, for a closed 1-form ρ on M, and φ is exact if and only if ρ is exact. ✷
1
A consideration of the stronger condition that the vector space H (M,R) = 0
yields the following corollary.
COROLLARY 7. Ricci equivalence and gauge equivalence are identical if and
1
only if H (M,R) = 0.
Since the Poincaré Lemma shows that the one-dimensional de Rham coho-
mology vector space of a contractible manifold is 0 (Conlon, 1993), Corollary 7
implies Ricci equivalence and gauge equivalence are identical when the base man-
ifold is contractible. In particular, it was noted in the previous section that all
TW-connections are Ricci equivalent when the base manifold has dimension one.
Therefore, if the base manifold has dimension one and is contractible, Corollary 7
shows that all TW-connections are gauge equivalent.
References
Bleecker, D. (1981) Gauge Theory and Variational Principles, Addison-Wesley, Reading, MA.
Conlon, L. (1993) Differentiable Manifolds: A First Course, Birkhäuser, Boston.
Kobayashi, S. (1972) Transformation Groups in Differential Geometry, Springer-Verlag, New York.
Roberts, C. W. (1992) The projective connections of T. Y. Thomas and J. H. C. Whitehead on the
principal R-bundle of volume elements, PhD Thesis, Saint Louis University, St. Louis, MO.
Roberts, C. W. (1995) The projective connections of T. Y. Thomas and J. H. C. Whitehead applied to
invariant connections, Differential Geom. Appl. 5, 237–255.
Spivak, M. (1979) A Comprehensive Introduction to Differential Geometry II, 2nd edn, Publish or
Perish, Wilmington.
Thomas, T. Y. (1925) On the projective and equi-projective geometries of paths, Proc. Nat. Acad.
Sci. 11, 199–203.
Thomas, T. Y. (1926) A projective theory of affinely connected manifolds, Math. Z. 25, 723–733.
Whitehead, J. H. C. (1931) The representation of projective spaces, Ann. of Math. 32, 327–360.
Mathematical Physics, Analysis and Geometry 7: 9–46, 2004.
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© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Heat Kernel Asymptotics of Zaremba Boundary
Value Problem
IVAN G. AVRAMIDI
Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro,
NM 87801, U.S.A. e-mail:
10 IVAN G. AVRAMIDI
sical smooth boundary problems (Dirichlet, Neumann, or a mixed combination of
those on vector bundles) are the most extensively studied ones (see [12, 13, 34, 2]
and the references therein). A more general scheme, so called oblique (or Grubb–
Gilkey–Smith) boundary value problem [31, 29, 28], which includes tangential
(oblique) derivatives along the boundary, has been studied in [7, 8, 6, 22–24]. In
this case the problem is not automatically elliptic; there is a certain strong ellipticity
condition on the leading symbol of the boundary operator. This problem is much
more difficult to handle, the main reason being that the heat kernel asymptotics
are no longer polynomial in the jets of the symbols of the differential operator
and the boundary operator. Another class of boundary value problems are charac-
terized by essentially nonlocal boundary conditions, for example, the spectral or
Atiyah–Patodi–Singer boundary conditions [30, 32, 11, 38].
All the boundary value problems described above are smooth. A more general
(and much more complicated) setting, so called singular boundary value problem,
arises when either the symbol of the differential operator or the symbol of the
boundary operator (or the boundary itself) are not smooth. In this paper we study
a singular boundary value problem for a second order partial differential operator
of Laplace type when the operator itself has smooth coefficients but the boundary
operator is not smooth. The case when the manifold as well as the boundary are
smooth, but the boundary operator jumps from Dirichlet type to Neumann type
along the boundary, is known in the literature as Zaremba problem. Such problems
often arise in applied mathematics and engineering and there are some exact re-
sults available for special cases (two or three dimensions, specific geometry, etc.)
[42, 25]. Zaremba problem belongs to a much wider class of singular boundary
value problems, i.e. manifolds with singularities (corners, edges, cones, etc.). There
is a large body of literature on this subject where the problem is studied from an
abstract function-analytical point of view [26, 14–18, 41, 37, 35, 33, 27]. However,
the study of heat kernel asymptotics of Zaremba type problems is quite new, and
there are only some preliminary results in this area [5, 40, 21, 20]. Moreover,
compared to the smooth category the needed machinery is still underdeveloped.
We would like to stress that we are interested not only in the asymptotics of the
trace of the heat kernel, i.e. the integrated heat kernel diagonal, but also in the local
asymptotic expansion of the off-diagonal heat kernel.
In this paper we study Zaremba boundary value problem for second-order par-
tial differential operators F of Laplace type acting on sections of a vector bundle
V over a smooth compact manifold M of dimension m with the boundary ∂M.
The boundary is decomposed as the disjoint union ∂M = 1 ∪ 2 ∪ 0, so
that 1 = 1 ∪ 0 and 2 = 2 ∪ 0 are smooth compact co-dimension one
submanifolds with the boundary 0 = ∂ 1 = ∂ 2, which is a smooth compact
co-dimension two submanifold without boundary, ∂ 0 = ∅. Both the manifold M
and its boundary ∂M are assumed to be smooth and the differential operator F
to have smooth coefficients. However, the boundary operator B is discontinuous
on the boundary, it jumps from the Dirichlet type operator on 1 to Neumann