Until very recently, all occurrences of American titanotheres have been known from either Eocene or lower Oligocene horizons of the Rocky Mountains and western Great Plains. Absence of any trace of these mammals in Tertiary deposits located in the region to the west of the Cordilleran Province lends considerable interest to the presence of titanotheres in one of the Sespe horizons of southern California

Stock, Chester

English

Caltech Authors - Main

GEOLOGY: C. STOCK
Crew, F. A. E., "Waved, an Autosomal Recessive Coat-Form Character in the Mouse,"
Jour. Genetics, 27, (1933).
Dunn, L. C., "Linkage in Rats and Mice," Genetics, 5 (1920).
Ibsen, H. L., "Linkage in Rats," Am. Nat., 54 (1916).
King, H. D., "Mutations in a Strain of Captive Gray Rats," Proc. Sixth Inter. Cong.
Genetics, 2, 50 (1932).
King, H. D., and Castle, W. E., "Linkage Studies of the Rat," Proc. Nat. Acad.
Sci., 390-399 (1935).
TITANOTHERE REMAINS FROM THE SESPE OF CALIFORNIA
By CHESTER STOCK
BALCH GRADUATE SCHOOL OF THE GEOLOGICAL SCIENCES, CALIFORNIA INSTITUTE OF
TECHNOLOGY
Communicated June 13, 1935
Introduction.-Until very recently, all occurrences of American titano-
theres have been known from either Eocene or lower Oligocene horizons
of the Rocky Mountains and western Great Plains. Absence of any trace
of these mammals in Tertiary deposits located in the region to the west
of the Cordilleran Province lends considerable interest to the presence of
titanotheres in one of the Sespe horizons of southern California.
Remains of titanotheres at Locality 150 have been recognized for several
years, since some of this material appeared early in the collection obtained
during the quarrying operations at that locality. Brief mention of the
occurrence of these mammals was made in preceding papers,' but a more
extended statement was reserved in the hope that further collecting might
reveal complete skull remains. Unfortunately, this hope has not been
realized. The desire to refer to the group in a broad survey of the Sespe
faunas and the recent discovery of titanothere material elsewhere in the
far west, make it urgent at this time to attempt an identification of the
type or types from the Sespe.
The rather numerous teeth and several jaws and jaw fragments, as
well as foot material, of titanotheres occurring at Locality 150 contrasts
markedly with the absence of such material in the collections obtained at
the upper Eocene localities situated lower in the Sespe section. It sub-
stantiates a field observation that the Brontotheriidae make their first
appearance or at least undergo their first noteworthy development in
Sespe time during the stage represented at Locality 150.
It is a matter of some interest, particukLrly from the standpoint of
establishing the relationship of the Sespe uppermost Eocene stage to the
Duchesne River horizon of Utah, that the titanotheres occurring in the
former are similar to those found in the latter. Indeed, so close is the
456 PROC. N. A. S.
CDC
r DO
o o
31 CD
t l
Htt,e
00;a,
r
-CD
Xo
CD1
0
A
0
P
0DS
PD
o)
GEOLOG Y: C. STOCK
resemblance between the two that there appear at present relatively few
and apparently minor characters separating them specifically.
Teleodus californicus, n.sp.
Type Specimen.-Fragmentary left ramus with lower dentition, No.
1398, C.I.T. Vert. Pale. Coll., plate 1, figure 1.
Paratype.-Anterior portion of lower jaw, No. 1120.
Referred Specimens.-A third incomplete lower jaw, No. 1126, and
miscellaneous lower and upper teeth, for the most part detached specimens,
some of which are shown in plates 1 and 2.
Locality.-AII the material referred to this titanothere comes from
Locality 150 in the Brea Canyon section of the Sespe, north of the Simi
Valley, California.
Characters.2-3 -, -, - . Incisors with rounded, non-cingulate crowns.
2-3 1 4 3
Third lower incisor absent in some individuals. Canines with crowns
considerably reduced in size. Upper canines with crowns not noticeably
enlarged at base. Very short diastema between C and P1.
P1 small; crown distinctly longer than wide with two external cusps,
an internal cingulum, and with a cusp-like enlargement of the inner
cingulum opposite forward portion of metacone (tritocone).
P2 rectangular in cross-section with transverse diameter greater than
anteroposterior. External surface convex on outer side of paracone and
metacone. Hypocone (tetartocone) joined with protocone (deutero-
cone), but constriction of intervening low ridge in at least one specimen
gives greater distinctness to the two cusps. Internal cingulum present
or absent.
P3 with outer face of paracone convex and with outer face of metacone
flatter. Deep pit present adjacent to ectoloph. Hypocone (tetartocone)
distinctly smaller than protocone (deuterocone) and well defined from
latter cusp. Internal cingulum always absent for short distance at base
of protocone; sometimes absent at base of hypocone.
M2 and M3 with anteroposterior diameter usually greater than trans-
DEscRIPTION OF PLATE 2
Teleodus californicus, n. sp.
Figure 1, M2 and M3, No. 1004; figure 2, P4, No. 1011; figure 3, M3, No. 1833;
figure 4, M3, No. 1095; figures 5 and 6, M3, Nos. 1835, 1836; figure 7, Ml and M2,
Nos. 1094, 1009; occlusal views; X 2/3.
Figure 8, lower canine, No. 1831, lateral view; figures 9, 9a, 10, lOa, incisors, Nos.
1838, 1837, side and inner views; figure 11, P3, No. 1091, occlusal view; X 2/3.
California Institute of Technology Collections. Sespe Uppermost Eocene, California.
458 PROC. N. A. S.
GEOLOG Y: C. STOCK4
2
7
.41
ga
PLATE 2
(Description on opposite page.)
VOL. 21, 1935 459
460 GEOLOGY: C. STOCK PROC. N. A. S.
verse diameter. Pits form adjacent to ectoloph and hypocone is small.
Hypocone in some specimens representing M3 with one or more subsidiary
cuspules on adjacent cingulum.
P1, small.
M3 may have well-developed posterior cingulum on posterior lobe.
The Sespe titanotheres are as small as or smaller than Teleodus uintensis.
T. californicus is distinguished principally by a precocious reduction of
the lower incisors from three to two in some individuals.
Comparisons.-Reviewing the numerous types of titanotheres described
from the Eocene and Oligocene of North America, the following genera,
because of their geologic position, need to be especially considered with
regard to possible relationship to the Sespe species, namely (1) Pro-
titanotherium, (2) Diplacodon, (3) Eotitanotherium, (4) Ateleodon and
(5) Teleodus. - Of these, the first three are definitely Eocene in age, while
the last two are from the Lower and Basal Oligocene.
(1) Protitanotherium differs from the Sespe form in decidedly larger
size, larger size of canines, and particularly in the larger size and more
primitive construction of the incisor teeth. A longer post-canine diastema
is present in the skull. PI and P2, relative to size of posterior premolars,
are larger than the comparable teeth in the Sespe species.
(2) Diplkwodon exhibits a number of progressive characters. In this
respect it shows some resemblance to the Sespe type. Thus, for example,
the premolars possess well developed tetartocones and rudimentary fos-
settes are present on the crown surface; the premolar tritocones are large
and are subequal with the protocones, and the molars have very distinct
fossettes or pits on the crown surface near the ectoloph.
Further resemblance to the Sespe form is seen in the elongate character
of the molars.
However, in the Sespe species there is a shorter diastema behind the
upper canine; P2 has acquired a rectangular cross-section and is thus
more advanced than the comparable tooth in Diplacodon; the premolars
usually do not have complete internal cingula; and the incisors (if we
may make comparison with these teeth in Peterson's species Diplacodon
superbum) have the more advanced rounded crowns. Moreover, the
lower incisors have undergone more reduction.
(3) Eotitanotherium is more progressive than Dipkacodon in the char-
acters displayed by the premolars and makes a closer approach to the
Sespe form in this regard.
However, like Protitanotherium, Eotitanotherium is a large type, con-
siderably larger than the Sespe species. Moreover, as in Protitanotherium,
the incisors are of more primitive construction. The post-canine diastema
in the skull is distinctly longer and the molars do not show the antero-
posterior elongation seen in Diplacodon and in the Sespe form.
GEOLOGY: C. STOCK
The three preceding types are from Eocene horizons definitely earlier
in time than that represented by Locality 150.
(4) Ateleodon, according to Schlaikjer,2 shows great reduction or even
loss of the lower canines and incisors. In this regard the Yoder genus
exhibits an advance beyond the stage represented by the Sespe specimens.
In the latter the incisors and canines are present, although these teeth
were on occasion small in size. The Sespe form resembles Ateleodon in
size. It is unfortunate that the incompleteness of the material from the
Yoder horizon limits severely the characterization of Ateleodon.
(5) Teleodus, or more particularly the species, T. uintensis, described
by Peterson3 from the Duchesne River, makes a close approach in its
characters to the Sespe form.
Some specimens from the Sespe are smaller, some slightly larger than
comparable materials from the Duchesne River, but on the whole there
is considerable similarity in character of size between the forms from
California and Utah. This difference among individuals from Locality
150 evidently expresses variation within a specific group and doubtless
is due also to a difference in sex. Thus far no structural differences appear
to furnish a basis for recognizing more than one specific type at the Cali-
fornia locality.
Compared with skull No. 11754 Carnegie Mus., the maxillary fragment
with C, P1 and P2, No. 1834 C.I.T., plate 1, figures 2, 2a, is similar in size
with P1 slightly more reduced. Two lower incisors are present in the
type, No. 1398, plate 1, figure 1. A distinct difference in size prevails
between these teeth, for the lateral, presumably second, incisor is con-
siderably larger than the medial one. In the larger jaw of the paratype,
No. 1120, the third incisor is likewise absent. However, in a third jaw from
the Sespe, No. 1126, remnants of the forward alveolar wall and external
septum suggest the presence of a small, laterally compressed alveolus for
I3. Unfortunately, in No. 1126, the crown of the canine immediately
adjacent has been pushed into the alveolus in such a way as to damage
considerably that portion of the jaw lying between the canine and I2.
This has prohibited a very clear demonstration of the presence of a third
incisor, but there nevertheless remains, on the basis of the evidence at
hand, considerable probability that 13 was present. In absence or reduced
size of 13, the Sespe species differs from Teleodus uintensis. The char-
acters displayed by the incisors in the former are certainly more like those
in Oligocene titanotheres than like those in Eocene types.
Plate 2 illustrates a number of teeth representing the upper and lower
dentitions of titanotheres from Locality 150. Individually, these speci-
mens resemble comparable teeth in T. uintensis.
1 Stock, C., Proc. Nat. Acad. Sci., 18, 522 (1932); Ibid., 19, 762 (1933).
VOL. 21, 1935 461
MA THEMA TICS: H. WHITNE Y
2 Schlaikjer, E. M., Mus. Comp. Zool. Bull., 76, 83-84 (1935).
Peterson, 0. A., Ann. Carnegie Mus., 20, art. 14 (1931).
DIFFERENTIABLE MANIFOLDS IN EUCLIDEAN SPACE'
BY HASSLER WHITNEY
DEPARTMEN.T OF MATHEMATICS, HARVARD UNIVERSITY
Communicated June 12, 1935
We present here a summary of some theorems on the imbedding of
abstract differentiable manifolds in Euclidean space E' and on the ap-
proximation to such manifolds by analytic manifolds. As a corollary it
is noted that any differentiable manifold may be given an analytic Rie-
mannian metric.
I. The Imbedding of a Differentiable Manifold in Euclidean Space.
Let M be a topological space with neighborhoods U1, U2, .... Let each
Ui be homeomorphic with the interior of the unit m-sphere S'. If U1
and Uj have common points Uij, then the homeomorphisms of Ui and Uj
with S' induce a mapping of one part of S' on another part. If all such
maps are of class Cr (i.e., have continuous partial derivatives through the
rth order), r _ 1, with non-vanishing Jacobian, we say M is differen-
tiable, and of class CT.
If M is in E' and each point of M is in a neighborhood which may be
defined by expressing n - m of the co6rdinates in terms of the remaining
m, the functions being of class Cr, then M is of class Cr in the above sense;
we say M is of class Cr in E". Suppose M is of class CT, and is mapped
into E". The n coordinates at points of M are n functions defined over
M. If these functions are of class CS (with the obvious definition for
s < r), and are independent (so that m independent directions at any
point of M go into m independent directions in E'), we call the map of
M in E' a regular Cr-map. Such a map is locally one-one: a neighborhood
of any point of M is mapped in a one-one manner in Et.
THEOREM I. *Any m-manifold of class CT (r > 1 finite or infinite) may
be imbedded by a regular CT-map in E2", and by su'h a map in a one-one
manner in E2m+i.
The proof runs as follows: If M is closed, a finite number of neighbor-
hoods U1, ..., U, cover M. Corresponding to these neighborhoods we
define functionsfi, .. ., f,,, Iu = (m + 1)v, of class Ct over M, which, used
as coordinates, map M in a regular Cr-manner in El. If r = 1, we next
approximate to M by a manifold of class C2. We now project M or the
new manifold along straight lines into spaces of lower dimension, till we
462 PROC. N. A. S.


Titanothere remains from the Sespe of California

GEOLOGY: C. STOCK
Crew, F. A. E., "Waved, an Autosomal Recessive Coat-Form Character in the Mouse,"
Jour. Genetics, 27, (1933).
Dunn, L. C., "Linkage in Rats and Mice," Genetics, 5 (1920).
Ibsen, H. L., "Linkage in Rats," Am. Nat., 54 (1916).
King, H. D., "Mutations in a Strain of Captive Gray Rats," Proc. Sixth Inter. Cong.
Genetics, 2, 50 (1932).
King, H. D., and Castle, W. E., "Linkage Studies of the Rat," Proc. Nat. Acad.
Sci., 390-399 (1935).
TITANOTHERE REMAINS FROM THE SESPE OF CALIFORNIA
By CHESTER STOCK
BALCH GRADUATE SCHOOL OF THE GEOLOGICAL SCIENCES, CALIFORNIA INSTITUTE OF
TECHNOLOGY
Communicated June 13, 1935
Introduction.-Until very recently, all occurrences of American titano-
theres have been known from either Eocene or lower Oligocene horizons
of the Rocky Mountains and western Great Plains. Absence of any trace
of these mammals in Tertiary deposits located in the region to the west
of the Cordilleran Province lends considerable interest to the presence of
titanotheres in one of the Sespe horizons of southern California.
Remains of titanotheres at Locality 150 have been recognized for several
years, since some of this material appeared early in the collection obtained
during the quarrying operations at that locality. Brief mention of the
occurrence of these mammals was made in preceding papers,' but a more
extended statement was reserved in the hope that further collecting might
reveal complete skull remains. Unfortunately, this hope has not been
realized. The desire to refer to the group in a broad survey of the Sespe
faunas and the recent discovery of titanothere material elsewhere in the
far west, make it urgent at this time to attempt an identification of the
type or types from the Sespe.
The rather numerous teeth and several jaws and jaw fragments, as
well as foot material, of titanotheres occurring at Locality 150 contrasts
markedly with the absence of such material in the collections obtained at
the upper Eocene localities situated lower in the Sespe section. It sub-
stantiates a field observation that the Brontotheriidae make their first
appearance or at least undergo their first noteworthy development in
Sespe time during the stage represented at Locality 150.
It is a matter of some interest, particukLrly from the standpoint of
establishing the relationship of the Sespe uppermost Eocene stage to the
Duchesne River horizon of Utah, that the titanotheres occurring in the
former are similar to those found in the latter. Indeed, so close is the
456 PROC. N. A. S.
CDC
r D
O
o o
31 CD
t l
Htt,e
00;a,
r
-CD
Xo
CD1
0
A
0
P
0DS
PD
o)
GEOLOG Y: C. STOCK
resemblance between the two that there appear at present relatively few
and apparently minor characters separating them specifically.
Teleodus californicus, n.sp.
Type Specimen.-Fragmentary left ramus with lower dentition, No.
1398, C.I.T. Vert. Pale. Coll., plate 1, figure 1.
Paratype.-Anterior portion of lower jaw, No. 1120.
Referred Specimens.-A third incomplete lower jaw, No. 1126, and
miscellaneous lower and upper teeth, for the most part detached specimens,
some of which are shown in plates 1 and 2.
Locality.-AII the material referred to this titanothere comes from
Locality 150 in the Brea Canyon section of the Sespe, north of the Simi
Valley, California.
Characters.2-3 -, -, - . Incisors with rounded, non-cingulate crowns.
2-3 1 4 3
Third lower incisor absent in some individuals. Canines with crowns
considerably reduced in size. Upper canines with crowns not noticeably
enlarged at base. Very short diastema between C and P1.
P1 small; crown distinctly longer than wide with two external cusps,
an internal cingulum, and with a cusp-like enlargement of the inner
cingulum opposite forward portion of metacone (tritocone).
P2 rectangular in cross-section with transverse diameter greater than
anteroposterior. External surface convex on outer side of paracone and
metacone. Hypocone (tetartocone) joined with protocone (deutero-
cone), but constriction of intervening low ridge in at least one specimen
gives greater distinctness to the two cusps. Internal cingulum present
or absent.
P3 with outer face of paracone convex and with outer face of metacone
flatter. Deep pit present adjacent to ectoloph. Hypocone (tetartocone)
distinctly smaller than protocone (deuterocone) and well defined from
latter cusp. Internal cingulum always absent for short distance at base
of protocone; sometimes absent at base of hypocone.
M2 and M3 with anteroposterior diameter usually greater than trans-
DEscRIPTION OF PLATE 2
Teleodus californicus, n. sp.
Figure 1, M2 and M3, No. 1004; figure 2, P4, No. 1011; figure 3, M3, No. 1833;
figure 4, M3, No. 1095; figures 5 and 6, M3, Nos. 1835, 1836; figure 7, Ml and M2,
Nos. 1094, 1009; occlusal views; X 2/3.
Figure 8, lower canine, No. 1831, lateral view; figures 9, 9a, 10, lOa, incisors, Nos.
1838, 1837, side and inner views; figure 11, P3, No. 1091, occlusal view; X 2/3.
California Institute of Technology Collections. Sespe Uppermost Eocene, California.
458 PROC. N. A. S.
GEOLOG Y: C. STOCK4
2
7
.41
ga
PLATE 2
(Description on opposite page.)
VOL. 21, 1935 459
460 GEOLOGY: C. STOCK PROC. N. A. S.
verse diameter. Pits form adjacent to ectoloph and hypocone is small.
Hypocone in some specimens representing M3 with one or more subsidiary
cuspules on adjacent cingulum.
P1, small.
M3 may have well-developed posterior cingulum on posterior lobe.
The Sespe titanotheres are as small as or smaller than Teleodus uintensis.
T. californicus is distinguished principally by a precocious reduction of
the lower incisors from three to two in some individuals.
Comparisons.-Reviewing the numerous types of titanotheres described
from the Eocene and Oligocene of North America, the following genera,
because of their geologic position, need to be especially considered with
regard to possible relationship to the Sespe species, namely (1) Pro-
titanotherium, (2) Diplacodon, (3) Eotitanotherium, (4) Ateleodon and
(5) Teleodus. - Of these, the first three are definitely Eocene in age, while
the last two are from the Lower and Basal Oligocene.
(1) Protitanotherium differs from the Sespe form in decidedly larger
size, larger size of canines, and particularly in the larger size and more
primitive construction of the incisor teeth. A longer post-canine diastema
is present in the skull. PI and P2, relative to size of posterior premolars,
are larger than the comparable teeth in the Sespe species.
(2) Diplkwodon exhibits a number of progressive characters. In this
respect it shows some resemblance to the Sespe type. Thus, for example,
the premolars possess well developed tetartocones and rudimentary fos-
settes are present on the crown surface; the premolar tritocones are large
and are subequal with the protocones, and the molars have very distinct
fossettes or pits on the crown surface near the ectoloph.
Further resemblance to the Sespe form is seen in the elongate character
of the molars.
However, in the Sespe species there is a shorter diastema behind the
upper canine; P2 has acquired a rectangular cross-section and is thus
more advanced than the comparable tooth in Diplacodon; the premolars
usually do not have complete internal cingula; and the incisors (if we
may make comparison with these teeth in Peterson's species Diplacodon
superbum) have the more advanced rounded crowns. Moreover, the
lower incisors have undergone more reduction.
(3) Eotitanotherium is more progressive than Dipkacodon in the char-
acters displayed by the premolars and makes a closer approach to the
Sespe form in this regard.
However, like Protitanotherium, Eotitanotherium is a large type, con-
siderably larger than the Sespe species. Moreover, as in Protitanotherium,
the incisors are of more primitive construction. The post-canine diastema
in the skull is distinctly longer and the molars do not show the antero-
posterior elongation seen in Diplacodon and in the Sespe form.
GEOLOGY: C. STOCK
The three preceding types are from Eocene horizons definitely earlier
in time than that represented by Locality 150.
(4) Ateleodon, according to Schlaikjer,2 shows great reduction or even
loss of the lower canines and incisors. In this regard the Yoder genus
exhibits an advance beyond the stage represented by the Sespe specimens.
In the latter the incisors and canines are present, although these teeth
were on occasion small in size. The Sespe form resembles Ateleodon in
size. It is unfortunate that the incompleteness of the material from the
Yoder horizon limits severely the characterization of Ateleodon.
(5) Teleodus, or more particularly the species, T. uintensis, described
by Peterson3 from the Duchesne River, makes a close approach in its
characters to the Sespe form.
Some specimens from the Sespe are smaller, some slightly larger than
comparable materials from the Duchesne River, but on the whole there
is considerable similarity in character of size between the forms from
California and Utah. This difference among individuals from Locality
150 evidently expresses variation within a specific group and doubtless
is due also to a difference in sex. Thus far no structural differences appear
to furnish a basis for recognizing more than one specific type at the Cali-
fornia locality.
Compared with skull No. 11754 Carnegie Mus., the maxillary fragment
with C, P1 and P2, No. 1834 C.I.T., plate 1, figures 2, 2a, is similar in size
with P1 slightly more reduced. Two lower incisors are present in the
type, No. 1398, plate 1, figure 1. A distinct difference in size prevails
between these teeth, for the lateral, presumably second, incisor is con-
siderably larger than the medial one. In the larger jaw of the paratype,
No. 1120, the third incisor is likewise absent. However, in a third jaw from
the Sespe, No. 1126, remnants of the forward alveolar wall and external
septum suggest the presence of a small, laterally compressed alveolus for
I3. Unfortunately, in No. 1126, the crown of the canine immediately
adjacent has been pushed into the alveolus in such a way as to damage
considerably that portion of the jaw lying between the canine and I2.
This has prohibited a very clear demonstration of the presence of a third
incisor, but there nevertheless remains, on the basis of the evidence at
hand, considerable probability that 13 was present. In absence or reduced
size of 13, the Sespe species differs from Teleodus uintensis. The char-
acters displayed by the incisors in the former are certainly more like those
in Oligocene titanotheres than like those in Eocene types.
Plate 2 illustrates a number of teeth representing the upper and lower
dentitions of titanotheres from Locality 150. Individually, these speci-
mens resemble comparable teeth in T. uintensis.
1 Stock, C., Proc. Nat. Acad. Sci., 18, 522 (1932); Ibid., 19, 762 (1933).
VOL. 21, 1935 461
MA THEMA TICS: H. WHITNE Y
2 Schlaikjer, E. M., Mus. Comp. Zool. Bull., 76, 83-84 (1935).
Peterson, 0. A., Ann. Carnegie Mus., 20, art. 14 (1931).
DIFFERENTIABLE MANIFOLDS IN EUCLIDEAN SPACE'
BY HASSLER WHITNEY
DEPARTMEN.T OF MATHEMATICS, HARVARD UNIVERSITY
Communicated June 12, 1935
We present here a summary of some theorems on the imbedding of
abstract differentiable manifolds in Euclidean space E' and on the ap-
proximation to such manifolds by analytic manifolds. As a corollary it
is noted that any differentiable manifold may be given an analytic Rie-
mannian metric.
I. The Imbedding of a Differentiable Manifold in Euclidean Space.
Let M be a topological space with neighborhoods U1, U2, .... Let each
Ui be homeomorphic with the interior of the unit m-sphere S'. If U1
and Uj have common points Uij, then the homeomorphisms of Ui and Uj
with S' induce a mapping of one part of S' on another part. If all such
maps are of class Cr (i.e., have continuous partial derivatives through the
rth order), r _ 1, with non-vanishing Jacobian, we say M is differen-
tiable, and of class CT.
If M is in E' and each point of M is in a neighborhood which may be
defined by expressing n - m of the co6rdinates in terms of the remaining
m, the functions being of class Cr, then M is of class Cr in the above sense;
we say M is of class Cr in E". Suppose M is of class CT, and is mapped
into E". The n coordinates at points of M are n functions defined over
M. If these functions are of class CS (with the obvious definition for
s < r), and are independent (so that m independent directions at any
point of M go into m independent directions in E'), we call the map of
M in E' a regular Cr-map. Such a map is locally one-one: a neighborhood
of any point of M is mapped in a one-one manner in Et.
THEOREM I. *Any m-manifold of class CT (r > 1 finite or infinite) may
be imbedded by a regular CT-map in E2", and by su'h a map in a one-one
manner in E2m+i.
The proof runs as follows: If M is closed, a finite number of neighbor-
hoods U1, ..., U, cover M. Corresponding to these neighborhoods we
define functionsfi, .. ., f,,, Iu = (m + 1)v, of class Ct over M, which, used
as coordinates, map M in a regular Cr-manner in El. If r = 1, we next
approximate to M by a manifold of class C2. We now project M or the
new manifold along straight lines into spaces of lower dimension, till we
462 PROC. N. A. S.


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Titanothere remains from the Sespe of California

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